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arXiv · 0709.3144

Inclusion Matrices and Chains

E. Ghorbani·G. B. Khosrovshahi·Ch. Maysoori·M. Mohammad-Noori
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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 $W_{tk}(v)$.

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BibTeXRIS

E. Ghorbani, G. B. Khosrovshahi, Ch. Maysoori, M. Mohammad-Noori. 2007-09-20. Inclusion Matrices and Chains. https://arxiv.org/abs/0709.3144

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