TOPICS IN ALGEBRAIC COMBINATORICS Richard P. Stanley Version of 1 February 2013 4 CONTENTS Preface 3 Notation 6 Chapter 1 Walks in graphs 9 Chapter 2 CubesandtheRadontransform 21 Chapter 3 Random walks 33 Chapter 4 The Sperner property 45 Chapter 5 Groupactionsonbooleanalgebras 59 Chapter 6 Young diagrams and q-binomial coefficients 77 Chapter 7 Enumeration under group action 99 Chapter 8 A glimpse of Young tableaux 131 Appendix The RSK algorithm 144 Appendix Plane partitions 147 Chapter 9 The Matrix-Tree Theorem 169 Appendix Three elegant combinatorial proofs 180 Chapter 10 Eulerian digraphs and oriented trees 189 Chapter 11 Cycles, bonds, and electrical networks 203 11.1 The cycle space and bond space 203 11.2 Bases for the cycle space and bond space 209 11.3 Electrical networks 214 11.4 Planar graphs (sketch) 220 11.5 Squaring the square 223 Chapter 12 Miscellaneous gems of algebraic combinatorics 231 12.1 The 100 prisoners 231 12.2 Oddtown 233 5 12.3 Complete bipartite partitions of Kn 234 12.4 The nonuniform Fisher inequality 236 12.5 Odd neighborhood covers 238 12.6 Circulant Hadamard matrices 240 12.7 P -recursive functions 246 Hints 257 References 261 6 Basic Notation P positive integers N nonnegative integers Z integers Q rational numbers R real numbers C complex numbers [n] the set 1, 2,...,n for n N (so [0] = ) { } ∈ ∅ Zn the group of integers modulo n R[x] the ring of polynomials in the variable x with coefficients in the ring R Y X for sets X and Y , the set of all functions f : X Y → := equal by definition Fq the finite field with q elements 2 j 1 (j) 1+ q + q + + q − ··· #S or S cardinality (number of elements) of the finite set S | | S T the disjoint union of S and T , i.e., S T , where S T = ∪· ∪ ∩ ∅ 2S the set of all subsets of the set S S k the set of k-element subsets of S S k the set of k-element multisets on S KS the vector space with basis S over the field K 7 Bn the poset of all subsets of [n], ordered by inclusion ρ(x) the rank of the element x in a graded poset [xn]F (x) coefficient of xn in the polynomial or power series F (x) x ⋖ y, y ⋗ x y covers x in a poset P δij the Kronecker delta, which equals 1 if i = j and 0 otherwise L the sum of the parts (entries) of L, if L is any array of | | nonnegative integers ℓ(λ) length (number of parts) of the partition λ p(n) number of partitions of the integer n 0 ≥ ker ϕ the kernel of a linear transformation or group homomorphism Sn symmetric group of all permutations of 1, 2,...,n ι the identity permutation of a set X, i.e., ι(x)= x for all x X ∈ 8 Chapter 1 Walks in graphs S Given a finite set S and integer k 0, let k denote the set of k-element subsets of S. A multiset may be regarded,≥ somewhat informally, as a set with repeated elements, such as 1, 1, 3, 4, 4, 4, 6, 6 . We are only concerned { } with how many times each element occurs, and not on any ordering of the elements. Thus for instance 2, 1, 2, 4, 1, 2 and 1, 1, 2, 2, 2, 4 are the same { } { } multiset: they each contain two 1’s, three 2’s, and one 4 (and no other elements). We say that a multiset M is on a set S if every element of M belongs to S. Thus the multiset in the example above is on the set S S = 1, 3, 4, 6 and also on any set containing S. Let k denote the set of{k-element} multisets on S. For instance, if S = 1, 2, 3 then (using { } abbreviated notation), S S = 12, 13, 23 , = 11, 22, 33, 12, 13, 23 . 2 { } 2 { } We now define what is meant by a graph. Intuitively, graphs have vertices and edges, where each edge “connects” two vertices (which may be the same). It is possible for two different edges e and e′ to connect the same two vertices. We want to be able to distinguish between these two edges, necessitating the following more precise definition. A (finite) graph G consists of a vertex set V = v1,...,vp and edge set E = e1,...,eq , together with a function { V } { } ϕ: E 2 . We think that if ϕ(e)= uv (short for u, v ), then e connects u and→v or equivalently e is incident to u and v. If there{ is} at least one edge incident to u and v then we say that the vertices u and v are adjacent. If ϕ(e) = vv, then we call e a loop at v. If several edges e1,...,ej (j > 1) satisfy ϕ(e ) = = ϕ(e ) = uv, then we say that there is a multiple edge 1 ··· j 9 10 CHAPTER 1. WALKS IN GRAPHS between u and v. A graph without loops or multiple edges is called simple. V In this case we can think of E as just a subset of 2 [why?]. The adjacency matrix of the graph G is the p p matrix A = A(G), over × the field of complex numbers, whose (i, j)-entry aij is equal to the number of edges incident to vi and vj. Thus A is a real symmetric matrix (and hence has real eigenvalues) whose trace is the number of loops in G. For instance, if G is the graph 1 2 3 4 5 then 21020 10001 A(G)= 00000 . 20001 01011 A walk in G of length ℓ from vertex u to vertex v is a sequence v1, e1, v2, e2,... , vℓ, eℓ, vℓ+1 such that: each v is a vertex of G • i each e is an edge of G • j the vertices of e are v and v , for 1 i ℓ • i i i+1 ≤ ≤ v = u and v = v. • 1 ℓ+1 1.1 Theorem. For any integer ℓ 1, the (i, j)-entry of the matrix A(G)ℓ ≥ is equal to the number of walks from vi to vj in G of length ℓ. 11 Proof. This is an immediate consequence of the definition of matrix multi- ℓ plication. Let A =(aij). The (i, j)-entry of A(G) is given by (A(G)ℓ) = a a a , ij ii1 i1i2 ··· iℓ−1j X where the sum ranges over all sequences (i1,...,iℓ 1) with 1 ik p. − ≤ ≤ But since ars is the number of edges between vr and vs, it follows that the summand a a a in the above sum is just the number (which may ii1 i1i2 ··· iℓ−1j be 0) of walks of length ℓ from vi to vj of the form vi, e1, vi1 , e2,...,viℓ−1 , eℓ, vj (since there are aii1 choices for e1, ai1i2 choices for e2, etc.) Hence summing over all (i1,...,iℓ 1) just gives the total number of walks of length ℓ from vi − to vj, as desired. We wish to use Theorem 1.1 to obtain an explicit formula for the number ℓ (A(G) )ij of walks of length ℓ in G from vi to vj. The formula we give will depend on the eigenvalues of A(G). The eigenvalues of A(G) are also called simply the eigenvalues of G. Recall that a real symmetric p p matrix M has p linearly independent real eigenvectors, which can in fact× be chosen to be orthonormal (i.e., orthogonal and of unit length). Let u1,...,up be real or- thonormal eigenvectors for M, with corresponding eigenvalues λ1,...,λp. All vectors u will be regarded as p 1 column vectors, unless specified otherwise. We let t denote transpose, so u×t isa1 p row vector. Thus the dot (or scalar or inner) product of the vectors u and×v is given by utv (ordinary matrix mul- t tiplication). In particular, uiuj = δij (the Kronecker delta). Let U = (uij) be the matrix whose columns are u1,...,up, denoted U = [u1,...,up]. Thus U is an orthogonal matrix, so t u1 t 1 . U = U − = . , ut p t t the matrix whose rows are u1,...,up. Recall from linear algebra that the matrix U diagonalizes M, i.e., 1 U − MU = diag(λ1,...,λp), where diag(λ1,...,λp) denotes the diagonal matrix with diagonal entries λ1,...,λp (in that order). 12 CHAPTER 1. WALKS IN GRAPHS 1.2 Corollary. Given the graph G as above, fix the two vertices vi and vj. Let λ1,...,λp be the eigenvalues of the adjacency matrix A(G). Then there exist real numbers c ,...,c such that for all ℓ 1, we have 1 p ≥ (A(G)ℓ) = c λℓ + + c λℓ . (1.1) ij 1 1 ··· p p 1 In fact, if U =(urs) is a real orthogonal matrix such that U − AU = diag(λ1,...,λp), then we have ck = uikujk. Proof. We have [why?] 1 ℓ ℓ ℓ U − A U = diag(λ1,...,λp). Hence ℓ ℓ ℓ 1 A = U diag(λ ,...,λ )U − . · 1 p 1 t Taking the (i, j)-entry of both sides (and using U − = U ) gives [why?] ℓ ℓ (A )ij = uikλkujk, Xk as desired. In order for Corollary 1.2 to be of any use we must be able to compute the eigenvalues λ1,...,λp as well as the diagonalizing matrix U (or eigenvectors ui).