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Paper IPM / M / 8742  


Abstract:  
Given integers t, k, and v such that 0 ≤ t ≤ k ≤ v, let W_{tk}(v) be the inclusion matrix of tsubsets vs. ksubsets of a vset. 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_{―tk}(v), which is rowequivalent 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}x=b due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of W_{―tk}(v) which is in some way equivalent to W_{tk}(v).
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