The story of the symmetric group

Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all The story of the symmetric group permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and Vipul Naik conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation March 20, 2007 map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Equivalently a permutation is a function from S to S for which we can find an inverse function.

The story of the What is a permutation? symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation Let S be a set. A permutation(defined) on S is a bijective The concept of cycle decomposition mapping S → S. Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the What is a permutation? symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation Let S be a set. A permutation(defined) on S is a bijective The concept of cycle decomposition mapping S → S. Cycle types and conjugacy classes Equivalently a permutation is a function from S to S for Two canonical maps which we can find an inverse function. Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? In particular, when S is a finite set, we can describe σ using the following two-line notation:

I The upper line lists the elements of S

I The lower line lists, under each element a ∈ S, the value σ(a).

The story of the How do we describe a permutation? symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I The upper line lists the elements of S

I The lower line lists, under each element a ∈ S, the value σ(a).

The story of the How do we describe a permutation? symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation In particular, when S is a finite set, we can describe σ using The concept of cycle decomposition the following two-line notation: Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the How do we describe a permutation? symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We can describe a permutation σ of a set S by providing a permutations rule that computes σ(a) for each a ∈ S. Cycle type of a permutation In particular, when S is a finite set, we can describe σ using The concept of cycle decomposition the following two-line notation: Cycle types and conjugacy classes Two canonical maps I The upper line lists the elements of S Block concatenations I The lower line lists, under each element a ∈ S, the General idea of a block concatenation value σ(a). map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? To prove this, first observe that the number of permutations of a set S is finite and is dependent only on the cardinality of S (and not on any additional structure with which S may be endowed). Further, observe that it is equal to the number of bijections between any two sets of the same cardinality. Call this number f (n). Now, pick a ∈ S. There are n possibilities for σ(a). Whatever value we choose for σ(a), there are n − 1 possible values for the images of the remaining elements under σ. Thus, for each choice of σ(a) there are f (n − 1) possibilities. Thus:

f (n) = nf (n − 1) And we have f (n) = n!

The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Further, observe that it is equal to the number of bijections between any two sets of the same cardinality. Call this number f (n). Now, pick a ∈ S. There are n possibilities for σ(a). Whatever value we choose for σ(a), there are n − 1 possible values for the images of the remaining elements under σ. Thus, for each choice of σ(a) there are f (n − 1) possibilities. Thus:

f (n) = nf (n − 1) And we have f (n) = n!

The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief To prove this, first observe that the number of permutations introduction to the symmetric group of a set S is finite and is dependent only on the cardinality The set of all permutations of S (and not on any additional structure with which S may The group of all be endowed). permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Now, pick a ∈ S. There are n possibilities for σ(a). Whatever value we choose for σ(a), there are n − 1 possible values for the images of the remaining elements under σ. Thus, for each choice of σ(a) there are f (n − 1) possibilities. Thus:

f (n) = nf (n − 1) And we have f (n) = n!

The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief To prove this, first observe that the number of permutations introduction to the symmetric group of a set S is finite and is dependent only on the cardinality The set of all permutations of S (and not on any additional structure with which S may The group of all be endowed). permutations Cycle type of a Further, observe that it is equal to the number of bijections permutation The concept of cycle decomposition between any two sets of the same cardinality. Call this Cycle types and conjugacy classes number f (n). Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the How many permutations are there? symmetric group Vipul Naik For a set S of size n, there are exactly n! permutations. A brief To prove this, first observe that the number of permutations introduction to the symmetric group of a set S is finite and is dependent only on the cardinality The set of all permutations of S (and not on any additional structure with which S may The group of all be endowed). permutations Cycle type of a Further, observe that it is equal to the number of bijections permutation The concept of cycle decomposition between any two sets of the same cardinality. Call this Cycle types and conjugacy classes number f (n). Two canonical maps Now, pick a ∈ S. There are n possibilities for σ(a). Block concatenations Whatever value we choose for σ(a), there are n − 1 possible General idea of a block concatenation values for the images of the remaining elements under σ. map For permutations Block concatenation Thus, for each choice of σ(a) there are f (n − 1) possibilities. on set-partitions Block concatenation Thus: on integer partitions Cardinality computations f (n) = nf (n − 1) Centralizers of permutations Classifying partitions, hence set-partitions And we have f (n) = n! and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? In this case, we can dispense with the two lines of the two-line notation for a permutation and just specify one line – the second line. The first line is understood to be {1, 2,..., n}.

The story of the Fix the set once and for all symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Since the structure of the set of permutations is the same for permutations Cycle type of a any set of size n, let us for convenience take the set of size n permutation The concept of cycle as {1, 2,..., n}. decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Fix the set once and for all symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Since the structure of the set of permutations is the same for permutations Cycle type of a any set of size n, let us for convenience take the set of size n permutation The concept of cycle as {1, 2,..., n}. decomposition Cycle types and In this case, we can dispense with the two lines of the conjugacy classes Two canonical maps two-line notation for a permutation and just specify one line Block – the second line. The first line is understood to be concatenations General idea of a block concatenation {1, 2,..., n}. map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Further, the composition operation is associative, hence we have an associative structure on the set of all permutations.

The story of the Composing as multiplication symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a Since permutations are functions, we can compose them as permutation The concept of cycle functions, and the composite of two permutations is a decomposition Cycle types and permutation. conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Composing as multiplication symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a Since permutations are functions, we can compose them as permutation The concept of cycle functions, and the composite of two permutations is a decomposition Cycle types and permutation. conjugacy classes Further, the composition operation is associative, hence we Two canonical maps Block have an associative structure on the set of all permutations. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Thus, we have on the set of permutations two structures:

I An associative multiplication given by function composition

I An identity map that is an identity for the associative multiplication

I An inverse map that is an inverse for this associative multiplication The set of permutations is now a group.

The story of the Inverses symmetric group Vipul Naik

A brief introduction to the By definition, a permutation is a function which has an symmetric group The set of all inverse function. Clearly, the inverse must itself be a permutations The group of all permutation. permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Inverses symmetric group Vipul Naik

A brief introduction to the By definition, a permutation is a function which has an symmetric group The set of all inverse function. Clearly, the inverse must itself be a permutations The group of all permutation. permutations Cycle type of a Thus, we have on the set of permutations two structures: permutation The concept of cycle decomposition I An associative multiplication given by function Cycle types and conjugacy classes composition Two canonical maps Block I An identity map that is an identity for the associative concatenations General idea of a multiplication block concatenation map For permutations I An inverse map that is an inverse for this associative Block concatenation on set-partitions multiplication Block concatenation on integer partitions The set of permutations is now a group. Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The symmetric group on {1, 2,..., n} is denoted as Sn and is termed the symmetric group on n letters.

The story of the Some notation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a For any set S, the group of all permutations of the set S is permutation The concept of cycle denoted as Sym(S) and is termed the symmetric decomposition Cycle types and group(defined) on S. conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Some notation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Cycle type of a For any set S, the group of all permutations of the set S is permutation The concept of cycle denoted as Sym(S) and is termed the symmetric decomposition Cycle types and group(defined) on S. conjugacy classes The symmetric group on {1, 2,..., n} is denoted as S and Two canonical maps n Block is termed the symmetric group on n letters. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? For u ∈ S, make an edge directed from u to f (u). The outdegree of every vertex is 1, and f is a permutation if and only if the indegree of every vertex is exactly one. But a directed graph with every vertex having indegree and outdegree 1, is simply a disjoint union of directed cycles.

The story of the A directed graph associated with a function symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Let S be a finite set and f : S → S be a function. Then, we permutations can associate to f a directed graph with vertex set S as Cycle type of a permutation follows. The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The outdegree of every vertex is 1, and f is a permutation if and only if the indegree of every vertex is exactly one. But a directed graph with every vertex having indegree and outdegree 1, is simply a disjoint union of directed cycles.

The story of the A directed graph associated with a function symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Let S be a finite set and f : S → S be a function. Then, we permutations can associate to f a directed graph with vertex set S as Cycle type of a permutation follows. The concept of cycle decomposition Cycle types and For u ∈ S, make an edge directed from u to f (u). conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? But a directed graph with every vertex having indegree and outdegree 1, is simply a disjoint union of directed cycles.

The story of the A directed graph associated with a function symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Let S be a finite set and f : S → S be a function. Then, we permutations can associate to f a directed graph with vertex set S as Cycle type of a permutation follows. The concept of cycle decomposition Cycle types and For u ∈ S, make an edge directed from u to f (u). conjugacy classes The outdegree of every vertex is 1, and f is a permutation if Two canonical maps Block and only if the indegree of every vertex is exactly one. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the A directed graph associated with a function symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Let S be a finite set and f : S → S be a function. Then, we permutations can associate to f a directed graph with vertex set S as Cycle type of a permutation follows. The concept of cycle decomposition Cycle types and For u ∈ S, make an edge directed from u to f (u). conjugacy classes The outdegree of every vertex is 1, and f is a permutation if Two canonical maps Block and only if the indegree of every vertex is exactly one. concatenations General idea of a But a directed graph with every vertex having indegree and block concatenation map outdegree 1, is simply a disjoint union of directed cycles. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? In other words, a cycle on S is a permutation where, for any a ∈ S, the set comprising {a, σ(a), σ2(a),..., } is the whole of S. The graph of a permutation is a disjoint union of cycles. Hence any permutation is a product of pairwise disjoint cycles (that is, cycles with no elements in common between any two).

The story of the Cycle decomposition of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A cycle is a permutation such that the graph associated with The group of all it is a cyclic graph. permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The graph of a permutation is a disjoint union of cycles. Hence any permutation is a product of pairwise disjoint cycles (that is, cycles with no elements in common between any two).

The story of the Cycle decomposition of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A cycle is a permutation such that the graph associated with The group of all it is a cyclic graph. permutations Cycle type of a In other words, a cycle on S is a permutation where, for any permutation The concept of cycle a ∈ S, the set comprising {a, σ(a), σ2(a),..., } is the whole decomposition Cycle types and conjugacy classes of S. Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Hence any permutation is a product of pairwise disjoint cycles (that is, cycles with no elements in common between any two).

The story of the Cycle decomposition of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A cycle is a permutation such that the graph associated with The group of all it is a cyclic graph. permutations Cycle type of a In other words, a cycle on S is a permutation where, for any permutation The concept of cycle a ∈ S, the set comprising {a, σ(a), σ2(a),..., } is the whole decomposition Cycle types and conjugacy classes of S. Two canonical maps The graph of a permutation is a disjoint union of cycles. Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Cycle decomposition of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A cycle is a permutation such that the graph associated with The group of all it is a cyclic graph. permutations Cycle type of a In other words, a cycle on S is a permutation where, for any permutation The concept of cycle a ∈ S, the set comprising {a, σ(a), σ2(a),..., } is the whole decomposition Cycle types and conjugacy classes of S. Two canonical maps The graph of a permutation is a disjoint union of cycles. Block concatenations Hence any permutation is a product of pairwise disjoint General idea of a block concatenation cycles (that is, cycles with no elements in common between map For permutations Block concatenation any two). on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The relation of having the same cycle type is an equivalence relation that partitions the set of all permutations into equivalence classes.

The story of the Cycle type of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all The cycle type of a permutation is the following data that permutations Cycle type of a can be inferred from the cycle decomposition. It is a permutation The concept of cycle sequence (i1, i2,...) where ij denotes the number of cycles of decomposition Cycle types and length j in the cycle decomposition of the given permutation. conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Cycle type of a permutation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all The cycle type of a permutation is the following data that permutations Cycle type of a can be inferred from the cycle decomposition. It is a permutation The concept of cycle sequence (i1, i2,...) where ij denotes the number of cycles of decomposition Cycle types and length j in the cycle decomposition of the given permutation. conjugacy classes Two canonical maps The relation of having the same cycle type is an equivalence Block concatenations relation that partitions the set of all permutations into General idea of a block concatenation equivalence classes. map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the The number of cycle types symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations Any cycle type (i1, i2,...) for permutations on a set of size n The group of all must satisfy the following: permutations Cycle type of a permutation X The concept of cycle ki = n decomposition k Cycle types and conjugacy classes k Two canonical maps In other words, cycle types correspond to unordered Block concatenations partitions of n. Thus, the number of cycle types for General idea of a block concatenation map permutations of length n is p(n), the number of partitions of For permutations Block concatenation the integer n. on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? When G = Sn, the conjugation map by g ∈ G has a particularly nice description. Namely cg (x) is the map that sends g(a) to g(x(a)). In other words, cg (x) is x twisted via a g-relabelling.

The story of the Conjugation as relabelling symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all In any group, we have certain inner automorphism defined permutations Cycle type of a via conjugation by an element. For g ∈ G, the conjugation permutation −1 The concept of cycle map by g is cg : x 7→ gxg . decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Conjugation as relabelling symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all In any group, we have certain inner automorphism defined permutations Cycle type of a via conjugation by an element. For g ∈ G, the conjugation permutation −1 The concept of cycle map by g is cg : x 7→ gxg . decomposition Cycle types and When G = Sn, the conjugation map by g ∈ G has a conjugacy classes Two canonical maps particularly nice description. Namely cg (x) is the map that Block sends g(a) to g(x(a)). concatenations General idea of a block concatenation In other words, cg (x) is x twisted via a g-relabelling. map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Conversely, given any two permutations of the same cycle type, there is a conjugation that takes one to the other. Hence two permutations have the same cycle type if and only if they are conjugate.

The story of the Conjugation preserves cycle type symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Since conjugation only relabels the elements, it does not permutations change the qualitative characteristics of the permutation. Cycle type of a permutation Thus, it sends each permutation to another permutation The concept of cycle decomposition Cycle types and with the same cycle type. conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Hence two permutations have the same cycle type if and only if they are conjugate.

The story of the Conjugation preserves cycle type symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Since conjugation only relabels the elements, it does not permutations change the qualitative characteristics of the permutation. Cycle type of a permutation Thus, it sends each permutation to another permutation The concept of cycle decomposition Cycle types and with the same cycle type. conjugacy classes Conversely, given any two permutations of the same cycle Two canonical maps Block type, there is a conjugation that takes one to the other. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Conjugation preserves cycle type symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Since conjugation only relabels the elements, it does not permutations change the qualitative characteristics of the permutation. Cycle type of a permutation Thus, it sends each permutation to another permutation The concept of cycle decomposition Cycle types and with the same cycle type. conjugacy classes Conversely, given any two permutations of the same cycle Two canonical maps Block type, there is a conjugation that takes one to the other. concatenations General idea of a Hence two permutations have the same cycle type if and block concatenation map only if they are conjugate. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The permutations of cycle type i = (i1, i2,...) form an orbit under the group’s action on itself by conjugation. Thus, the number of permutations is the cardinality of the group divided by the cardinality of the isotropy group of any one permutation having that cycle type.

The story of the Centralizers and conjugacy classes symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We now have a somewhat group-theoretic twist to the permutations problem of computing the number of permutations of a Cycle type of a permutation given cycle type. The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Thus, the number of permutations is the cardinality of the group divided by the cardinality of the isotropy group of any one permutation having that cycle type.

The story of the Centralizers and conjugacy classes symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We now have a somewhat group-theoretic twist to the permutations problem of computing the number of permutations of a Cycle type of a permutation given cycle type. The concept of cycle decomposition Cycle types and The permutations of cycle type i = (i1, i2,...) form an orbit conjugacy classes under the group’s action on itself by conjugation. Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Centralizers and conjugacy classes symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all We now have a somewhat group-theoretic twist to the permutations problem of computing the number of permutations of a Cycle type of a permutation given cycle type. The concept of cycle decomposition Cycle types and The permutations of cycle type i = (i1, i2,...) form an orbit conjugacy classes under the group’s action on itself by conjugation. Two canonical maps Block Thus, the number of permutations is the cardinality of the concatenations General idea of a group divided by the cardinality of the isotropy group of any block concatenation map one permutation having that cycle type. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I OB(n) is the set of all partitions of a set of n elements into an ordered sequence of sets.

I B(n) is the set of all equivalence relations (or set partitions) of a set of n elements. The cardinality of this set is Bn (the Bell number)

I OP(n) is the set of all ordered partitions of n into nonnegative integers

I P(n) is the set of all unordered partitions of n into nonnegative integers. The cardinality of this set is p(n)

The story of the Combinatorial structures of interest to us symmetric group Vipul Naik

A brief Five combinatorial structures of interest to us are: introduction to the symmetric group The set of all I Sn or Sym(n) is the set of all permutations on a set of permutations The group of all order n. The cardinality of this set is n! permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I B(n) is the set of all equivalence relations (or set partitions) of a set of n elements. The cardinality of this set is Bn (the Bell number)

I OP(n) is the set of all ordered partitions of n into nonnegative integers

I P(n) is the set of all unordered partitions of n into nonnegative integers. The cardinality of this set is p(n)

The story of the Combinatorial structures of interest to us symmetric group Vipul Naik

A brief Five combinatorial structures of interest to us are: introduction to the symmetric group The set of all I Sn or Sym(n) is the set of all permutations on a set of permutations The group of all order n. The cardinality of this set is n! permutations Cycle type of a I OB(n) is the set of all partitions of a set of n elements permutation The concept of cycle into an ordered sequence of sets. decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I OP(n) is the set of all ordered partitions of n into nonnegative integers

I P(n) is the set of all unordered partitions of n into nonnegative integers. The cardinality of this set is p(n)

The story of the Combinatorial structures of interest to us symmetric group Vipul Naik

A brief Five combinatorial structures of interest to us are: introduction to the symmetric group The set of all I Sn or Sym(n) is the set of all permutations on a set of permutations The group of all order n. The cardinality of this set is n! permutations Cycle type of a I OB(n) is the set of all partitions of a set of n elements permutation The concept of cycle into an ordered sequence of sets. decomposition Cycle types and conjugacy classes I B(n) is the set of all equivalence relations (or set Two canonical maps partitions) of a set of n elements. The cardinality of Block concatenations this set is Bn (the Bell number) General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I P(n) is the set of all unordered partitions of n into nonnegative integers. The cardinality of this set is p(n)

The story of the Combinatorial structures of interest to us symmetric group Vipul Naik

A brief Five combinatorial structures of interest to us are: introduction to the symmetric group The set of all I Sn or Sym(n) is the set of all permutations on a set of permutations The group of all order n. The cardinality of this set is n! permutations Cycle type of a I OB(n) is the set of all partitions of a set of n elements permutation The concept of cycle into an ordered sequence of sets. decomposition Cycle types and conjugacy classes I B(n) is the set of all equivalence relations (or set Two canonical maps partitions) of a set of n elements. The cardinality of Block concatenations this set is Bn (the Bell number) General idea of a block concatenation map I OP(n) is the set of all ordered partitions of n into For permutations Block concatenation nonnegative integers on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Combinatorial structures of interest to us symmetric group Vipul Naik

A brief Five combinatorial structures of interest to us are: introduction to the symmetric group The set of all I Sn or Sym(n) is the set of all permutations on a set of permutations The group of all order n. The cardinality of this set is n! permutations Cycle type of a I OB(n) is the set of all partitions of a set of n elements permutation The concept of cycle into an ordered sequence of sets. decomposition Cycle types and conjugacy classes I B(n) is the set of all equivalence relations (or set Two canonical maps partitions) of a set of n elements. The cardinality of Block concatenations this set is Bn (the Bell number) General idea of a block concatenation map I OP(n) is the set of all ordered partitions of n into For permutations Block concatenation nonnegative integers on set-partitions Block concatenation on integer partitions I P(n) is the set of all unordered partitions of n into Cardinality nonnegative integers. The cardinality of this set is p(n) computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Another way of putting this is that it gives the equivalence relation of being in the same connected component of the directed graph associated with σ. Note that this map is independent of the labels on the elements of the set, and thus, in particular, it is equivariant under inner automorphisms.

The story of the Canonical map from permutations to symmetric group set-partitions Vipul Naik

A brief introduction to the symmetric group The set of all There is a map: permutations The group of all permutations Cycle type of a Pr : Sn → B(n) permutation The concept of cycle decomposition that sends a permutation σ to the equivalence relation of Cycle types and conjugacy classes being in the same cycle in the cycle decomposition of σ. Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Canonical map from permutations to symmetric group set-partitions Vipul Naik

A brief introduction to the symmetric group The set of all There is a map: permutations The group of all permutations Cycle type of a Pr : Sn → B(n) permutation The concept of cycle decomposition that sends a permutation σ to the equivalence relation of Cycle types and conjugacy classes being in the same cycle in the cycle decomposition of σ. Two canonical maps Block Another way of putting this is that it gives the equivalence concatenations General idea of a relation of being in the same connected component of the block concatenation map directed graph associated with σ. For permutations Block concatenation Note that this map is independent of the labels on the on set-partitions Block concatenation elements of the set, and thus, in particular, it is equivariant on integer partitions Cardinality under inner automorphisms. computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This gives a map:

Upr : B(n) → P(n)

The composite CT = Upr ◦ Pr is the cycle type map that sends each permutation to its cycle type.

The story of the Canonical map from set-partitions to unordered symmetric group integer partitions Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given a set partition, the corresponding unordered integer permutations Cycle type of a partition is the partition that uses each integer j as many permutation The concept of cycle times as there are subsets of size j. decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The composite CT = Upr ◦ Pr is the cycle type map that sends each permutation to its cycle type.

The story of the Canonical map from set-partitions to unordered symmetric group integer partitions Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given a set partition, the corresponding unordered integer permutations Cycle type of a partition is the partition that uses each integer j as many permutation The concept of cycle times as there are subsets of size j. decomposition Cycle types and This gives a map: conjugacy classes Two canonical maps Block Upr : B(n) → P(n) concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Canonical map from set-partitions to unordered symmetric group integer partitions Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given a set partition, the corresponding unordered integer permutations Cycle type of a partition is the partition that uses each integer j as many permutation The concept of cycle times as there are subsets of size j. decomposition Cycle types and This gives a map: conjugacy classes Two canonical maps Block Upr : B(n) → P(n) concatenations General idea of a block concatenation map For permutations Block concatenation The composite CT = Upr ◦ Pr is the cycle type map that on set-partitions Block concatenation sends each permutation to its cycle type. on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Two questions of interest:

I Given π ∈ P(n), what can we say about the sizes of Upr −1(π) and Pr −1 ◦ Upr −1(π)?

I Suppose we construct a further map:

P(n) → {1, 2,..., n} What can we say about the inverse images of k ∈ {1, 2,..., n} via the composite mappings?

The story of the Cardinality questions that we are interested in symmetric group Vipul Naik

A brief We have the following maps: introduction to the symmetric group The set of all permutations Sn → B(n) → P(n) The group of all permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Cardinality questions that we are interested in symmetric group Vipul Naik

A brief We have the following maps: introduction to the symmetric group The set of all permutations Sn → B(n) → P(n) The group of all permutations Cycle type of a permutation Two questions of interest: The concept of cycle decomposition Cycle types and conjugacy classes I Given π ∈ P(n), what can we say about the sizes of Two canonical maps −1 −1 −1 Upr (π) and Pr ◦ Upr (π)? Block concatenations Suppose we construct a further map: General idea of a I block concatenation map For permutations Block concatenation P(n) → {1, 2,..., n} on set-partitions Block concatenation on integer partitions What can we say about the inverse images of Cardinality k ∈ {1, 2,..., n} via the composite mappings? computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the General notion of block concatenation symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Suppose we have a family of sets Tm for every integer permutations Cycle type of a m ∈ N. Then, a block concatenation structure on the Tms is permutation The concept of cycle the following kind of structure: decomposition Cycle types and For each m, n ∈ N, we have a map: conjugacy classes Two canonical maps Block Φm,n : Tm × Tn → Tm+n concatenations General idea of a block concatenation map Satisfying a certain associativity condition. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Block concatenation of more than two things symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given an ordered partition n = m1 + m2 + ... + mr , we have permutations Cycle type of a an injective homomorphism: permutation The concept of cycle decomposition Cycle types and conjugacy classes Φm1,m2,...,mr : Tm1 × Tm2 × ... × Tmr → Tn Two canonical maps This essentially keeps applying the block concatenation two Block concatenations at a time, and uses associativity to guarantee that the result General idea of a block concatenation map is well-defined. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This gives an injective homomorphism (which is a block concatenation):

Φm,n : Sm × Sn → Sm+n

The image of Φm,n is termed the Young subgroup for the ordered pair (m, n).

The story of the Block concatenation on permutations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Let m, n ∈ N, with g ∈ Sm and h ∈ Sn. Then, consider the permutations The group of all element of Sm+n defined as follows: It acts like g on the first permutations m letters and acts like h on the remaining n letters (treating Cycle type of a permutation m + k as k). The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Block concatenation on permutations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Let m, n ∈ N, with g ∈ Sm and h ∈ Sn. Then, consider the permutations The group of all element of Sm+n defined as follows: It acts like g on the first permutations m letters and acts like h on the remaining n letters (treating Cycle type of a permutation m + k as k). The concept of cycle decomposition Cycle types and This gives an injective homomorphism (which is a block conjugacy classes concatenation): Two canonical maps Block concatenations General idea of a Φm,n : Sm × Sn → Sm+n block concatenation map For permutations Block concatenation The image of Φm,n is termed the Young subgroup for the on set-partitions Block concatenation ordered pair (m, n). on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The Young subgroup can equivalently be thought of as those elements of the group of all permutations that stabilize each part in the partition.

The story of the Block concatenation of multiple permutations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Given an ordered integer partition n = m + m + ... + m , permutations 1 2 r The group of all we have a map: permutations Cycle type of a permutation The concept of cycle decomposition Φm1,m2,...,mr : Sm1 × Sm2 × ... × Smr → Sn Cycle types and conjugacy classes The image of this map is termed the Young subgroup Two canonical maps Block corresponding to the ordered partition. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Block concatenation of multiple permutations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Given an ordered integer partition n = m + m + ... + m , permutations 1 2 r The group of all we have a map: permutations Cycle type of a permutation The concept of cycle decomposition Φm1,m2,...,mr : Sm1 × Sm2 × ... × Smr → Sn Cycle types and conjugacy classes The image of this map is termed the Young subgroup Two canonical maps Block corresponding to the ordered partition. concatenations General idea of a The Young subgroup can equivalently be thought of as those block concatenation map elements of the group of all permutations that stabilize each For permutations Block concatenation part in the partition. on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? that simply takes the union of the ordered set partitions for the subsets of sizes mi for each i.

The story of the For ordered set-partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations Let n = m + m + ... + m be an ordered integer partition The group of all 1 2 r permutations of n. Then, we have an injective map: Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Φm1,m2,...,mr : OB(m1) × OB(m2) × ... × OB(mr ) → OB(n) Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the For ordered set-partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations Let n = m + m + ... + m be an ordered integer partition The group of all 1 2 r permutations of n. Then, we have an injective map: Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Φm1,m2,...,mr : OB(m1) × OB(m2) × ... × OB(mr ) → OB(n) Two canonical maps Block concatenations General idea of a block concatenation that simply takes the union of the ordered set partitions for map For permutations the subsets of sizes mi for each i. Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? (The map continues to remain injective because, given any partition that arises as the image of a block concatenation, we can uniquely retrieve the set-partitions in each block that gave rise to it).

The story of the For unordered set-partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations We can also perform a block concatenation on unordered set Cycle type of a permutation partitions. This will again give an injective map. The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the For unordered set-partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations We can also perform a block concatenation on unordered set Cycle type of a permutation partitions. This will again give an injective map. (The map The concept of cycle decomposition continues to remain injective because, given any partition Cycle types and conjugacy classes that arises as the image of a block concatenation, we can Two canonical maps Block uniquely retrieve the set-partitions in each block that gave concatenations General idea of a rise to it). block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This map is again injective.

The story of the For ordered integer partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given an ordered integer partition n = m1 + m2 + ... mr , we permutations Cycle type of a have a map: permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Φm1,m2,...,mr : OP(m1)×OP(m2)×...×OP(m−r) → OP(n) Block concatenations General idea of a Which simply concatenates the partitions one after the other. block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the For ordered integer partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all Given an ordered integer partition n = m1 + m2 + ... mr , we permutations Cycle type of a have a map: permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Φm1,m2,...,mr : OP(m1)×OP(m2)×...×OP(m−r) → OP(n) Block concatenations General idea of a Which simply concatenates the partitions one after the other. block concatenation map This map is again injective. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Unlike all the previous maps this one is not injective. That is, there could be different unordered integer partition tuples that give rise to the same unordered integer partition under block concatenation.

The story of the For unordered integer partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Given an unordered integer partition n = m1 + m2 ... mr , we permutations The group of all have a map: permutations Cycle type of a permutation The concept of cycle decomposition Φm ,m ,...,mr : P(m1) × P(m2) × ... × P(mr ) → P(n) Cycle types and 1 2 conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the For unordered integer partitions symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all Given an unordered integer partition n = m1 + m2 ... mr , we permutations The group of all have a map: permutations Cycle type of a permutation The concept of cycle decomposition Φm ,m ,...,mr : P(m1) × P(m2) × ... × P(mr ) → P(n) Cycle types and 1 2 conjugacy classes Two canonical maps Block concatenations Unlike all the previous maps this one is not injective. That General idea of a block concatenation map is, there could be different unordered integer partition tuples For permutations Block concatenation that give rise to the same unordered integer partition under on set-partitions Block concatenation block concatenation. on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? We also have a weaker notion of symmetry for a partition. Namely, we are interested in those permutations that preserve the equivalence relation of the partition. In other words, we are interested in permutations that send each set of the partition completely to another set of the partition.

The story of the Stabilizer for a whole partition symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations Given an unordered set partition of the set {1, 2,..., n} we The group of all can compute the set of those permutations that fix every set permutations Cycle type of a in the partition. This, it turns out, is isomorphic to the permutation The concept of cycle decomposition product of symmetric groups for every part. Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Stabilizer for a whole partition symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations Given an unordered set partition of the set {1, 2,..., n} we The group of all can compute the set of those permutations that fix every set permutations Cycle type of a in the partition. This, it turns out, is isomorphic to the permutation The concept of cycle decomposition product of symmetric groups for every part. Cycle types and conjugacy classes We also have a weaker notion of symmetry for a partition. Two canonical maps Namely, we are interested in those permutations that Block concatenations preserve the equivalence relation of the partition. In other General idea of a block concatenation words, we are interested in permutations that send each set map For permutations Block concatenation of the partition completely to another set of the partition. on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Then the group of symmetries that preserve the equivalence relation induced by the partition is described as follows. It is a direct product of groups D(j, ij ) where D(j, ij ) is the group of permutations on the union of the blocks of size j that take each block to a block. It turns out that:

ij D(j, ij ) = (Sj ) o Sij And the symmetry group for a partition is: Y D(j, ij ) j

The story of the Cardinality of the symmetry group of a partition symmetric group Vipul Naik

Consider an unordered partition of n with ij occurrences of j A brief introduction to the for each j. symmetric group The set of all permutations The group of all permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? It turns out that:

ij D(j, ij ) = (Sj ) o Sij And the symmetry group for a partition is: Y D(j, ij ) j

The story of the Cardinality of the symmetry group of a partition symmetric group Vipul Naik

Consider an unordered partition of n with ij occurrences of j A brief introduction to the for each j. symmetric group The set of all Then the group of symmetries that preserve the equivalence permutations The group of all relation induced by the partition is described as follows. It is permutations Cycle type of a a direct product of groups D(j, ij ) where D(j, ij ) is the group permutation The concept of cycle of permutations on the union of the blocks of size j that decomposition Cycle types and take each block to a block. conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Cardinality of the symmetry group of a partition symmetric group Vipul Naik

Consider an unordered partition of n with ij occurrences of j A brief introduction to the for each j. symmetric group The set of all Then the group of symmetries that preserve the equivalence permutations The group of all relation induced by the partition is described as follows. It is permutations Cycle type of a a direct product of groups D(j, ij ) where D(j, ij ) is the group permutation The concept of cycle of permutations on the union of the blocks of size j that decomposition Cycle types and take each block to a block. conjugacy classes It turns out that: Two canonical maps Block concatenations General idea of a ij block concatenation D(j, ij ) = (Sj ) o Sij map For permutations Block concatenation And the symmetry group for a partition is: on set-partitions Block concatenation on integer partitions Y Cardinality D(j, ij ) computations j Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Description of the centralizer symmetric group Vipul Naik

A brief introduction to the symmetric group let σ be a permutation of cycle type (i , i ,...) The set of all 1 2 permutations The group of all Then: permutations Cycle type of a Y permutation CSn (σ) = C(j, ij ) The concept of cycle decomposition j Cycle types and conjugacy classes where: Two canonical maps Block concatenations General idea of a ij block concatenation C(j, ij ) = (Z/jZ) o Sij map For permutations Block concatenation this is a subgroup of the group of all symmetries of the on set-partitions Block concatenation partition. on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Some explicit formulae symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Let π = (i1, i2,...) be an unordered partition of n. Then, we Cycle type of a have: permutation The concept of cycle decomposition I The number of set-partitions corresponding to π is Cycle types and −1 conjugacy classes [Sn : D(j, ij )]. Recall that this is CT (π). Two canonical maps Block I The number of permutations corresponding to π is concatenations −1 General idea of a [Sn : C(j, ij )]. Recall that this is Pr (π). block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This essentially divides Sn or B(n) into n or n + 1 parts.

The story of the Recall of motivations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations We had maps: The group of all permutations Cycle type of a permutation Sn → B(n) → P(n) The concept of cycle decomposition Cycle types and We had observed that given any map k : P(n) → S that conjugacy classes Two canonical maps essentially “classifies” unordered integer partitions, we get Block corresponding maps Sn → S and B(n) → S. S here may concatenations General idea of a block concatenation usually be {1, 2,..., n} or {0, 1,..., n}. map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Recall of motivations symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations We had maps: The group of all permutations Cycle type of a permutation Sn → B(n) → P(n) The concept of cycle decomposition Cycle types and We had observed that given any map k : P(n) → S that conjugacy classes Two canonical maps essentially “classifies” unordered integer partitions, we get Block corresponding maps Sn → S and B(n) → S. S here may concatenations General idea of a block concatenation usually be {1, 2,..., n} or {0, 1,..., n}. map For permutations This essentially divides Sn or B(n) into n or n + 1 parts. Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I Permutations get classified according to the number of cycles. The number of permutations with k cycles is termed s0(n, k) and is called the unsigned Stirling number of the first kind. I Set partitions get classified according to the number of subsets. The number of set-partitions with k subsets is termed S(n, k) and is called the Stirling number of the second kind.

The story of the The total number of parts symmetric group One possibility for k is the total number of parts Vipul Naik

corresponding to the unordered integer partition, viz for an A brief introduction to the unordered integer partition (i!, i2,...), we define: symmetric group The set of all permutations X The group of all k = ij permutations j Cycle type of a permutation We can partition the set of partitions based on the value of The concept of cycle decomposition Cycle types and k (k can range from 1 to n). conjugacy classes Corresponding to this, we get the following way of classifying Two canonical maps Block permutations: concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? I Set partitions get classified according to the number of subsets. The number of set-partitions with k subsets is termed S(n, k) and is called the Stirling number of the second kind.

The story of the The total number of parts symmetric group One possibility for k is the total number of parts Vipul Naik

corresponding to the unordered integer partition, viz for an A brief introduction to the unordered integer partition (i!, i2,...), we define: symmetric group The set of all permutations X The group of all k = ij permutations j Cycle type of a permutation We can partition the set of partitions based on the value of The concept of cycle decomposition Cycle types and k (k can range from 1 to n). conjugacy classes Corresponding to this, we get the following way of classifying Two canonical maps Block permutations: concatenations General idea of a block concatenation I Permutations get classified according to the number of map For permutations cycles. The number of permutations with k cycles is Block concatenation 0 on set-partitions termed s (n, k) and is called the unsigned Stirling Block concatenation on integer partitions number of the first kind. Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the The total number of parts symmetric group One possibility for k is the total number of parts Vipul Naik

corresponding to the unordered integer partition, viz for an A brief introduction to the unordered integer partition (i!, i2,...), we define: symmetric group The set of all permutations X The group of all k = ij permutations j Cycle type of a permutation We can partition the set of partitions based on the value of The concept of cycle decomposition Cycle types and k (k can range from 1 to n). conjugacy classes Corresponding to this, we get the following way of classifying Two canonical maps Block permutations: concatenations General idea of a block concatenation I Permutations get classified according to the number of map For permutations cycles. The number of permutations with k cycles is Block concatenation 0 on set-partitions termed s (n, k) and is called the unsigned Stirling Block concatenation on integer partitions number of the first kind. Cardinality computations I Set partitions get classified according to the number of Centralizers of permutations subsets. The number of set-partitions with k subsets is Classifying partitions, hence set-partitions termed S(n, k) and is called the Stirling number of the and permutations Concatenation- second kind. invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? As per this classification:

I The number of permutations with k fixed points is n
k Dn−k where Dr is the number of permutations on r elements with no fixed point.

I The number of unordered set partitions with k fixed n
points is k B2(n − k) where B2(r) is the number of unordered set partitions with every set having size at least 2.

The story of the The number of singleton parts symmetric group Vipul Naik

A brief introduction to the Another approach is to define k = i1, that is, we classify symmetric group The set of all unordered integer partitions of n according to the number of permutations The group of all times 1 occurs in the integer partition. permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the The number of singleton parts symmetric group Vipul Naik

A brief introduction to the Another approach is to define k = i1, that is, we classify symmetric group The set of all unordered integer partitions of n according to the number of permutations The group of all times 1 occurs in the integer partition. permutations Cycle type of a As per this classification: permutation The concept of cycle decomposition I The number of permutations with k fixed points is Cycle types and conjugacy classes n
Two canonical maps k Dn−k where Dr is the number of permutations on r Block elements with no fixed point. concatenations General idea of a I The number of unordered set partitions with k fixed block concatenation n map points is
B (n − k) where B (r) is the number of For permutations k 2 2 Block concatenation on set-partitions unordered set partitions with every set having size at Block concatenation least 2. on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? One way of looking at the above statement is as follows: For any set S of size n, we can define:

Sym(S) → B(S) → P(n) Now, Sym(S) acts on S, and hence also on B(S) and on Sym(S) (by inner automorphism). The claim is that the map from Sym(S) to B(S) is equivariant under Sym(S) action, and the right map is invariant under Sym(S) action.

The story of the What we have seen so far symmetric group Vipul Naik

A brief introduction to the So far, the various structural elements we have seen are symmetric group The set of all unchanged upto relabelling. In other words, if we altered the permutations The group of all labels, the maps would also simply get relabelled. The maps permutations Cycle type of a do not fundamentally depend on the order of the elements. permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the What we have seen so far symmetric group Vipul Naik

A brief introduction to the So far, the various structural elements we have seen are symmetric group The set of all unchanged upto relabelling. In other words, if we altered the permutations The group of all labels, the maps would also simply get relabelled. The maps permutations Cycle type of a do not fundamentally depend on the order of the elements. permutation The concept of cycle One way of looking at the above statement is as follows: decomposition Cycle types and For any set S of size n, we can define: conjugacy classes Two canonical maps Block Sym(S) → B(S) → P(n) concatenations General idea of a block concatenation map Now, Sym(S) acts on S, and hence also on B(S) and on For permutations Block concatenation Sym(S) (by inner automorphism). The claim is that the on set-partitions Block concatenation map from Sym(S) to B(S) is equivariant under Sym(S) on integer partitions Cardinality action, and the right map is invariant under Sym(S) action. computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Why this is restrictive symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Looking only at structures that are invariant under Cycle type of a permutation conjugation or relabelling causes us to miss out on a lot of The concept of cycle decomposition fun. For instance, we cannot try to see whether 1 occurs Cycle types and conjugacy classes before 2, whether 2 occurs before 3, how the numbers rise Two canonical maps Block and fall etc. Thus, we’d ideally like to look for structures concatenations General idea of a and maps that can exploit the total ordering of {1, 2,..., n}. block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Why this is restrictive symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Looking only at structures that are invariant under Cycle type of a permutation conjugation or relabelling causes us to miss out on a lot of The concept of cycle decomposition fun. For instance, we cannot try to see whether 1 occurs Cycle types and conjugacy classes before 2, whether 2 occurs before 3, how the numbers rise Two canonical maps Block and fall etc. Thus, we’d ideally like to look for structures concatenations General idea of a and maps that can exploit the total ordering of {1, 2,..., n}. block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Concatenation-invariance: what it means symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations Suppose we have two sequences An and Dn of sets, both Cycle type of a equipped with block concatenation maps, and a map permutation The concept of cycle fn : An → Dn for each n. decomposition Cycle types and We say that f s are compatible with the concatenation (or conjugacy classes n Two canonical maps concatenation-invariant) if concatenating before applying the Block concatenations fns has the same effect as concatenating after applying the General idea of a block concatenation fns. map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Why we would like concatenation-invariance symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations The group of all permutations The broad reason is that we want that if something (a Cycle type of a permutation permutation, a set-partition, or whatever) does not mix up The concept of cycle decomposition the first m elements with the last n elements, its behaviour Cycle types and conjugacy classes on the first m elements should be like something for just m Two canonical maps Block elements and its behaviour for the last n elements should be concatenations General idea of a just like for n elements. block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This gives a map:

M1 : Sn → OB(n) And incidentally, also a map to OP(n).

The story of the Some motivation symmetric group Vipul Naik

Recall that we have a natural map: A brief introduction to the symmetric group The set of all Upr : Sn → B(n) permutations The group of all permutations Which takes a permutation and outputs the set partition Cycle type of a induced by its cycle decomposition. permutation The concept of cycle decomposition Intrinsically, this set partition is unordered. However, if we Cycle types and conjugacy classes think of the underlying set as {1, 2,..., n}, we can order the Two canonical maps parts by their least elements. That is, for disjoint subsets A Block concatenations and B of {1, 2,..., n} say that A is less than B if the least General idea of a block concatenation map element of A is less than the least element of B. For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Some motivation symmetric group Vipul Naik

Recall that we have a natural map: A brief introduction to the symmetric group The set of all Upr : Sn → B(n) permutations The group of all permutations Which takes a permutation and outputs the set partition Cycle type of a induced by its cycle decomposition. permutation The concept of cycle decomposition Intrinsically, this set partition is unordered. However, if we Cycle types and conjugacy classes think of the underlying set as {1, 2,..., n}, we can order the Two canonical maps parts by their least elements. That is, for disjoint subsets A Block concatenations and B of {1, 2,..., n} say that A is less than B if the least General idea of a block concatenation map element of A is less than the least element of B. For permutations Block concatenation This gives a map: on set-partitions Block concatenation on integer partitions Cardinality M1 : Sn → OB(n) computations Centralizers of permutations And incidentally, also a map to OP(n). Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? It now turns out that we can define a bijection B : Sn → Sn such that B ◦ M1 = M2. In other words, we can take a permutation and write another permutation such that the one-line notation of one interprets the cycle decomposition of the other. This map is concatenation-invariant.

The story of the Another map symmetric group Vipul Naik

A brief We now consider another map: introduction to the symmetric group The set of all permutations The group of all M2 : Sn → OB(n) permutations Cycle type of a And thus, also a map to OP(n). permutation The concept of cycle That takes a given permutation, writes it in one-line decomposition Cycle types and notation, and then breaks the permutation at each conjugacy classes Two canonical maps left-to-right minimum. Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? This map is concatenation-invariant.

The story of the Another map symmetric group Vipul Naik

A brief We now consider another map: introduction to the symmetric group The set of all permutations The group of all M2 : Sn → OB(n) permutations Cycle type of a And thus, also a map to OP(n). permutation The concept of cycle That takes a given permutation, writes it in one-line decomposition Cycle types and notation, and then breaks the permutation at each conjugacy classes Two canonical maps left-to-right minimum. Block concatenations It now turns out that we can define a bijection B : Sn → Sn General idea of a block concatenation such that B ◦ M1 = M2. map For permutations In other words, we can take a permutation and write another Block concatenation on set-partitions permutation such that the one-line notation of one interprets Block concatenation on integer partitions the cycle decomposition of the other. Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Another map symmetric group Vipul Naik

A brief We now consider another map: introduction to the symmetric group The set of all permutations The group of all M2 : Sn → OB(n) permutations Cycle type of a And thus, also a map to OP(n). permutation The concept of cycle That takes a given permutation, writes it in one-line decomposition Cycle types and notation, and then breaks the permutation at each conjugacy classes Two canonical maps left-to-right minimum. Block concatenations It now turns out that we can define a bijection B : Sn → Sn General idea of a block concatenation such that B ◦ M1 = M2. map For permutations In other words, we can take a permutation and write another Block concatenation on set-partitions permutation such that the one-line notation of one interprets Block concatenation on integer partitions the cycle decomposition of the other. Cardinality computations This map is concatenation-invariant. Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? We define a map:

A : Sn → OB(n) That sends a permutation to the ordered set partitions that break the permutations at the points where it falls. Thus, each part in the ordered partition is a rising run in the permutation. This also gives a map:

0 A : Sn → OP(n) And another:

00 A : Sn → P(n)

The story of the Yet another map! symmetric group The map described above takes a permutation and breaks it Vipul Naik

at the left-to-right cumulative minima. Another interesting A brief introduction to the map is one where we try to see whether the value is rising or symmetric group The set of all falling in the immediate sense. permutations The group of all permutations Cycle type of a permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Yet another map! symmetric group The map described above takes a permutation and breaks it Vipul Naik

at the left-to-right cumulative minima. Another interesting A brief introduction to the map is one where we try to see whether the value is rising or symmetric group The set of all falling in the immediate sense. permutations The group of all We define a map: permutations Cycle type of a permutation A : Sn → OB(n) The concept of cycle decomposition Cycle types and That sends a permutation to the ordered set partitions that conjugacy classes Two canonical maps break the permutations at the points where it falls. Thus, Block each part in the ordered partition is a rising run in the concatenations General idea of a block concatenation permutation. map For permutations This also gives a map: Block concatenation on set-partitions Block concatenation 0 on integer partitions A : Sn → OP(n) Cardinality computations And another: Centralizers of permutations Classifying partitions, hence set-partitions 00 and permutations A : Sn → P(n) Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? P The sum j ij gives the total number of rising runs. We thus have a map:

000 A : Sn → {1, 2,..., n} The size of the inverse image of a given k is termed the Eulerian number A(n, k).

The story of the The Eulerian numbers symmetric group Vipul Naik

A brief introduction to the We have a map: symmetric group The set of all permutations The group of all 00 permutations A : Sn → P(n) Cycle type of a permutation which counts the number ij of rising runs in a give The concept of cycle decomposition permutation of every length j. Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? We thus have a map:

000 A : Sn → {1, 2,..., n} The size of the inverse image of a given k is termed the Eulerian number A(n, k).

The story of the The Eulerian numbers symmetric group Vipul Naik

A brief introduction to the We have a map: symmetric group The set of all permutations The group of all 00 permutations A : Sn → P(n) Cycle type of a permutation which counts the number ij of rising runs in a give The concept of cycle decomposition permutation of every length j. Cycle types and conjugacy classes P Two canonical maps The sum j ij gives the total number of rising runs. Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the The Eulerian numbers symmetric group Vipul Naik

A brief introduction to the We have a map: symmetric group The set of all permutations The group of all 00 permutations A : Sn → P(n) Cycle type of a permutation which counts the number ij of rising runs in a give The concept of cycle decomposition permutation of every length j. Cycle types and conjugacy classes P Two canonical maps The sum j ij gives the total number of rising runs. Block We thus have a map: concatenations General idea of a block concatenation 000 map A : Sn → {1, 2,..., n} For permutations Block concatenation on set-partitions Block concatenation The size of the inverse image of a given k is termed the on integer partitions Eulerian number A(n, k). Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the Outline symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all permutations A brief The group of all permutations introduction to the symmetric group The set of all Cycle type of a permutation permutations The concept of cycle decomposition The group of all Cycle types and conjugacy classes permutations Two canonical maps Cycle type of a permutation Block concatenations The concept of cycle General idea of a block concatenation map decomposition For permutations Cycle types and Block concatenation on set-partitions conjugacy classes Block concatenation on integer partitions Two canonical maps Block Cardinality computations concatenations Centralizers of permutations General idea of a block concatenation Classifying partitions, hence set-partitions and permutations map For permutations Concatenation-invariant structures Block concatenation Conjugation-invariant structure on set-partitions Block concatenation Concatenation-invariance on integer partitions The leader representation The Eulerian numbers Cardinality computations Centralizers of Young tableaux permutations So what’s next? Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The maps we considered initially depended only on the cycle decomposition: a conjugation-invariant structure. The maps involving left-to-right minima and rises and falls used the order structure. However, even the maps that have so far used the order, have used it very ineffectively. They haven’t exploited much of the information inherent in a permutation.

The story of the A close look at what’s been so far symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all So far, we have looked at various maps from Sn to sets such permutations The group of all as B(n), P(n) and {1, 2,..., n} (often, with one map permutations Cycle type of a factoring through the other). permutation The concept of cycle decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? However, even the maps that have so far used the order, have used it very ineffectively. They haven’t exploited much of the information inherent in a permutation.

The story of the A close look at what’s been so far symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all So far, we have looked at various maps from Sn to sets such permutations The group of all as B(n), P(n) and {1, 2,..., n} (often, with one map permutations Cycle type of a factoring through the other). permutation The concept of cycle The maps we considered initially depended only on the cycle decomposition Cycle types and decomposition: a conjugation-invariant structure. The maps conjugacy classes involving left-to-right minima and rises and falls used the Two canonical maps Block order structure. concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the A close look at what’s been so far symmetric group Vipul Naik

A brief introduction to the symmetric group The set of all So far, we have looked at various maps from Sn to sets such permutations The group of all as B(n), P(n) and {1, 2,..., n} (often, with one map permutations Cycle type of a factoring through the other). permutation The concept of cycle The maps we considered initially depended only on the cycle decomposition Cycle types and decomposition: a conjugation-invariant structure. The maps conjugacy classes involving left-to-right minima and rises and falls used the Two canonical maps Block order structure. concatenations General idea of a block concatenation However, even the maps that have so far used the order, map For permutations have used it very ineffectively. They haven’t exploited much Block concatenation on set-partitions of the information inherent in a permutation. Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The rising and falling runs approach was poor because it looked only at the local relationships, and it failed to capture the way entries far away linked with each other. Thus, if we want to preserve maximum information about a permutation, we have to store both the local behaviour (that with respect to the things just before) and a global behaviour (That with respect to all the things that have been encountered so far). The method for doing this is Young tableaux.

The story of the A two-dimensional information storage symmetric group mechanism Vipul Naik

A brief introduction to the The left-to-right minimum approach was poor in terms of symmetric group The set of all information storage because it neglected all the things that permutations The group of all were not left-to-right minima. For instance, for all permutations Cycle type of a permutations with the first entry 1, the left-to-right permutation The concept of cycle minimum information is the same. decomposition Cycle types and conjugacy classes Two canonical maps Block concatenations General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? Thus, if we want to preserve maximum information about a permutation, we have to store both the local behaviour (that with respect to the things just before) and a global behaviour (That with respect to all the things that have been encountered so far). The method for doing this is Young tableaux.

The story of the A two-dimensional information storage symmetric group mechanism Vipul Naik

A brief introduction to the The left-to-right minimum approach was poor in terms of symmetric group The set of all information storage because it neglected all the things that permutations The group of all were not left-to-right minima. For instance, for all permutations Cycle type of a permutations with the first entry 1, the left-to-right permutation The concept of cycle minimum information is the same. decomposition Cycle types and The rising and falling runs approach was poor because it conjugacy classes Two canonical maps looked only at the local relationships, and it failed to capture Block concatenations the way entries far away linked with each other. General idea of a block concatenation map For permutations Block concatenation on set-partitions Block concatenation on integer partitions Cardinality computations Centralizers of permutations Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next? The story of the A two-dimensional information storage symmetric group mechanism Vipul Naik

A brief introduction to the The left-to-right minimum approach was poor in terms of symmetric group The set of all information storage because it neglected all the things that permutations The group of all were not left-to-right minima. For instance, for all permutations Cycle type of a permutations with the first entry 1, the left-to-right permutation The concept of cycle minimum information is the same. decomposition Cycle types and The rising and falling runs approach was poor because it conjugacy classes Two canonical maps looked only at the local relationships, and it failed to capture Block concatenations the way entries far away linked with each other. General idea of a block concatenation Thus, if we want to preserve maximum information about a map For permutations permutation, we have to store both the local behaviour (that Block concatenation on set-partitions Block concatenation with respect to the things just before) and a global on integer partitions behaviour (That with respect to all the things that have Cardinality computations been encountered so far). Centralizers of permutations The method for doing this is Young tableaux. Classifying partitions, hence set-partitions and permutations Concatenation- invariant structures Conjugation-invariant structure Concatenation- invariance The leader representation The Eulerian numbers Young tableaux So what’s next?