Inclusion Matrices and Chains

TL;DR

This paper studies inclusion matrices of subsets, introduces a new matrix with similar properties, and provides a new proof for Wilson's theorem on integral solutions, using combinatorial decompositions and Smith normal forms.

Contribution

It constructs a new inclusion matrix row-equivalent to the standard one and determines its Smith normal form, offering new insights and proofs related to subset inclusion matrices.

Findings

Decomposition of the poset into symmetric skipless chains

Construction of a new inclusion matrix with similar properties

A new proof of Wilson's theorem on integral solutions

Abstract

Given integers $t$, $k$, and $v$ such that $0\leq t\leq k\leq v$, let $W_{tk}(v)$ 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 $W_{\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ 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 $W_{tk}\bf{x}=\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$.