Discrete Applied Mathematics 34 (1991) 241-277 241 North-Holland Some applications of algebra to combinatorics Richard P. Stanley * Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 20 August 1989 Revised 26 February 1990 Abstract Stanley, R.P., Some applications of algebra to combinatorics, Discrete Applied Mathematics 34 (1991) 241-277. In extremal combinatorics, it is often convenient to work in the context of partial- ly ordered sets. First let us establish some notation and definitions. As general references on the subject of partially ordered sets we recommend [I; 28, Chapter 31. 1. Definitions A partially ordered set (poset) is a set together with a binary relation which is reflexive, antisymmetric, and transitive. Let P be a (finite) graded poset, P=P,UP,U...UP,;PjistheithrankofP,andweletpi=\P,l bethenumberof elements of rank i. Every maximal chain of P passes through exactly one element of each of the subsets Pi, starting from rank 0, and going up through rank 1, then rank 2, etc. The posets we will consider will be graded and each maximal chain will have length n (that is, n + 1 elements). The rank generating function of P is the polynomial F(P, q) = Cy=, piq’y and it is a useful construct in the study of various properties of P. The poset properties in which we are interested here are: rank symmetry (i.e., pi=pn_i, for i= 0 to n), * Partially supported by NSF grant #DMS-8401376. I am grateful to Rodica Simion for providing me the opportunity to lecture at the Capital City Conference on Combinatorics and Theoretical Computer Science, and for her careful preparation of these notes based on my lectures. 0166-218X/91/$03.50 0 1991 - Elsevier Science Publishers B.V. All rights reserved 242 R.P. Stanley 123 Fig. 1. The boolean algebra B3 and an example of a non-Sperner poset. rank unimodality (i.e., there is j such that pOsp, 5 .-- ‘pj’pj+ I 2 -a. zp,), and the Spernerproperty (defined below). Of course, if a poset is both rank symmetric and rank unimodal, then the middle rank(s) achieve the maximum cardinality among all the ranks of P. An antichain is a subset A c P no two elements of which are comparable in P. Clearly, in a graded poset each rank Pi is an antichain and hence maxA IA 1~ maxipi, where A ranges over the antichains in P. If equality holds, then we say that the poset P is Sperner. Thus, in a Sperner poset the largest rank provides an antichain of maximum cardinality, but there may exist other antichains of max- imum cardinality as well. So, if a poset P is known to be rank symmetric, rank unimodal, and Sperner, then we know that the maximum cardinality of an antichain in P is the cardinality of its middle rank(s). Figure 1 shows a poset of rank 1 which is not Sperner (the largest pi equals 3 while the poset contains an antichain of car- dinality 4) and the boolean aIgebra Bs which is Sperner. Many interesting problems can be formulated in terms of the Sperner property of some poset. An important example is the boolean algebra B,, which is the poset of all subsets of an n-element set ordered by inclusion. In B, the rank of an ele- ment is given by its cardinality, and there are pi = (7) elements of rank i. We have the rank generating function F(B,, q) = (1 + q)“, and the rank symmetry and unimodality of B, are obvious from well-known properties of binomial coeffi- cients. It is not equally obvious whether B,, is Sperner. In fact, the origin of the ter- minology goes back to Emmanuel Sperner who proved in 1927 that: Sperner’s theorem 1.1. The boolean algebra B, is Sperner, for each n 2 1. Sperner’s theorem can be stated without reference to posets: given an n-element set, what is the maximum number of subsets you can select so that none of the subsets contains another? Sperner’s result says that one cannot exceed (tnn/21), which can be achieved by taking all the subsets of cardinality [n/2]. There are many refinements and generalizations of the Sperner property, but we will keep things simple by considering only the Sperner property. Some applications of algebra to combinatorics 243 Next, we will relate the Sperner property to matchings. A useful working condi- tion which implies that a rank unimodal poset P is Sperner is the existence of an order matching between any two consecutive levels. The map ,!I : Pi+ Pi+, , or /.I : Pi+, + Pi, is an order matching if ,u is one-to-one and ,LIrespects the order, i.e., ,u(x)>x, or p(x)<x, for all XE P. In connection with the Sperner property we have the following simple proposition. Proposition 1.2. Suppose that in the poset P there exist order matchings PO-+ P, + 1.. jPjtP. c ... e-p,. /+I Then P is rank unimodal and Sperner, with pi= lllaX;pi. Proof. The unimodality property is clear from the definition of an order matching. The order matchings between successive ranks give rise to a partition of P into (dis- joint) chains, each of which intersects Pj. Therefore, the number of chains is Pi. On the other hand, every antichain A intersects each chain in at most one point; hence, IA) Ipi, so P is Sperner. 0 Now we will bring algebra into the picture, starting with linear algebra, and later building more algebraic machinery. 2. Linear algebra Given a poset P, define QP to be the vector space over the field Q of rational numbers (any other field could be used), consisting of formal linear combinations of elements of P with rational coefficients. Assume now that P is graded. Note that QP is the direct sum of the subspaces spanned by the ranks of P, so we have QP=QPO@QP,@~~~@QP,,. If xEP, we let C+(x):={y~P:rank(y)=rank(x)+l and y>x}, that is, C’(x) denotes the set of elements in P which cover x. Similar- ly, we denote by C-(x) the set of elements which are covered by x, C-(x) := {y~P:rank(y)=rank(x)-1 andy<x}. The following is a key definition establishing a special kind of linear operator on the vector space QP in which we will be interested. A linear operator U: QP- QP is order raising if U(x) E QC’(x) for all XE P. Thus, U(x) is a linear combination of elements which cover x, and it is denoted U for up. We relate now the linear algebra with the Sperner property through the following proposition. Proposition 2.1. Let U: QPj --f QP;, 1 be (the restriction to the ith rank of) an order raising operator. If U is one-to-one, then there exists an order matching p:P;dP;+,. Note that this result gives us “more room to work” when we want to prove that a poset has an order matching. Instead of having to exhibit the actual matching of 244 R.P. Stanley elements from consecutive ranks, we only need to exhibit an order raising operator mapping elements to linear combinations of elements from the next rank. This is easier because the set of linear combinations is much larger than the rank itself, so there are many more possibilities for an order raising operator than for an order matching. Proof. Look at the matrix of the linear transformation U with respect to the bases P, and Pi+, . Thus, it is a pi by pi+, matrix whose rows are indexed by the elements in Pi= {Xl,X2, .._ ,x,,} and whose columns are indexed by the elements in Pi+, = {YI?Y,, .**3y,,,,}. Let r be the rank of this matrix. Since U is one-to-one, we have r=p;, and let us assume that the row and column indexing is such that the r by r minor formed by the first r columns is not zero. In particular, there must exist a nonzero term in the expansion of this determinant. Permute the rows if necessary so that the diagonal term of this minor is nonzero, that is, each diagonal entry in the matrix is nonzero. But, if the ith diagonal entry is nonzero, it means that when U is applied to xi, the element yi appears with nonzero coefficient in U(x,). Since U is order raising this means further that x1 is covered by y,, and that I =yi gives an order matching ,u: Pi- P;+, . Cl 2.1. Applications Let us first apply this result to the boolean algebra. Alternatively, an order mat- ching for B,, can be exhibited, by giving an explicit association of each element of B, of rank k, k<n/2, with a particular element which covers it (or which it covers, if k>n/2). This however is not so easy. In the case of our approach, it will suffice to map each element of B, of rank k, k<n/2, to a linear combination of all the elements which cover it. The linear algebra will do the work for us and supply an order matching, by ensuring that in the linear combinations associated with dif- ferent elements of rank k, a different element has nonzero coefficient. We will need to prove only that the linear transformation is one-to-one if k<n/2 and onto if km/2. Let us do the simplest thing and take U: QB, --$QB, defined by U(S)= c TEC+CSjT; that is, to each subset SEB, we associate the sum of all the subsets which cover it.