Inclusion matrices and chains

Given integers t , k, and v such that 0 t k v, let Wtk(v) be the inclusion matrix of t-subsets vs. k-subsets of a v-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset 2[v] into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by Wtk(v), which is row-equivalent to Wtk(v). Its Smith normal form is determined. As applications, Wilson’s diagonal form of Wtk(v) is obtained as well as a new proof of the well-known theorem on the necessary and sufficient conditions for existence of integral solutions of the system Wtkx = b due to Wilson. Finally we present another inclusion matrix with similar properties to those of Wtk(v) which is in some way equivalent to Wtk(v). © 2007 Elsevier Inc. All rights reserved.