TOPICS IN ALGEBRAIC COMBINATORICS Richard P. Stanley to Kenneth and Sharon 3 Preface This book is intended primarily as a one-semester undergraduate text for a course in algebraic combinatorics. The main prerequisites are a basic knowledge of linear algebra (eigenvalues, eigenvectors, etc.) over a field, ex- istence of finite fields, and some rudimentary understanding of group theory. The one exception is Section 12.6, which involves finite extensions of the ra- tionals including a little Galois theory. Prior knowledge of combinatorics is not essential but will be helpful. Why write an undergraduate textbook on algebraic combinatorics? One obvious reason is simply to gather some material that I find very interesting and hope that students will agree. A second reason concerns students who have taken an introductory algebra course and want to know what can be done with their new-found knowledge. Undergraduate courses that require a basic knowledge of algebra are typically either advanced algebra courses or abstract courses on subjects like algebraic topology and algebraic geometry. Algebraic combinatorics offers a byway off the traditional algebraic highway, one that is more intuitive and more easily accessible. Algebraic combinatorics is a huge subject, so some selection process was necessary to obtain the present text. The main results, such as the weak Erd˝os-Moser theorem and the enumeration of de Bruijn sequences, have the feature that their statement does not involve any algebra. Such results are good advertisements for the unifying power of algebra and for the unity of mathematics as a whole. All but the last chapter are vaguely connected to walks on graphs and linear transformations related to them. The final chapter is a hodgepodge of some unrelated elegant applications of algebra to combinatorics. The sections of this chapter are independent from each other and the rest of the text. There are also three chapter appendices on purely enumerational aspects of combinatorics related to the chapter mate- rial: the RSK algorithm, plane partitions, and the enumeration of labelled trees. Almost all the material covered here can serve as a gateway to much additional algebraic combinatorics. We hope in fact that this book will serve exactly this purpose, that is, to inspire its readers to delve more deeply into the fascinating interplay between algebra and combinatorics. 4 CONTENTS Preface 3 Notation 6 Chapter 1 Walks in graphs 9 Chapter 2 Cubes and the Radon transform 19 Chapter 3 Random walks 31 Chapter 4 The Sperner property 35 Chapter 5 Group actions on boolean algebras 49 Chapter 6 Young diagrams and q-binomial coefficients 65 Chapter 7 Enumeration under group action 87 Chapter 8 A glimpse of Young tableaux 119 Appendix The RSK algorithm 132 Appendix Plane partitions 135 Chapter 9 The Matrix-Tree Theorem 157 Appendix Three elegant combinatorial proofs 168 Chapter 10 Eulerian digraphs and oriented trees 177 Chapter 11 Cycles, bonds, and electrical networks 191 11.1 The cycle space and bond space 191 11.2 Bases for the cycle space and bond space 197 11.3 Electrical networks 202 11.4 Planar graphs (sketch) 208 11.5 Squaring the square 211 Chapter 12 Miscellaneous gems of algebraic combinatorics 219 12.1 The 100 prisoners 219 12.2 Oddtown 221 5 12.3 Complete bipartite partitions of Kn 222 12.4 The nonuniform Fisher inequality 224 12.5 Odd neighborhood covers 226 12.6 Circulant Hadamard matrices 228 12.7 P -recursive functions 234 Hints 245 References 249 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 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 = 1, 3, 4, 6 and also on any set containing S { } 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 { } µ ¶ µµ ¶¶ A (finite) graph G consists of a vertex set V = v1,...,vp and edge set { V } E = e1,...,eq , together with a function 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 e ,...,e (j > 1) satisfy ϕ(e )= = ϕ(e )= uv, 1 j 1 ··· j then we say that there is a multiple edge between u and v. A graph without loops or multiple edges is called simple. In this case we can think of E as V 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 9 10 CHAPTER 1. WALKS IN GRAPHS 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. 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 11 (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 orthonormal eigenvectors for M, with corresponding eigenvalues λ1, . , λp. All vectors u will be regarded as p 1 column vectors. We let t denote trans- pose, so ut is a 1 p row vector. Thus× the dot (or scalar or inner) product of the vectors u and×v is given by utv (ordinary matrix multiplication).