Inclusion matrices and chains

Given integers t, k, and v such that 0 ⩽ t ⩽ k ⩽ v , let W t k ( 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 W t ¯ k ( v ) , which is row-equivalent to W t k ( v ) . Its Smith normal form is determined. As applications, Wilsonʹs diagonal form of W t k ( 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 W t k x = b due to Wilson. Finally we present another inclusion matrix with similar properties to those of W t ¯ k ( v ) which is in some way equivalent to W t k ( v ) .