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Inclusion matrices and chains

Ghorbani, E ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1016/j.jcta.2007.09.002
  3. Publisher: 2008
  4. Abstract:
  5. Given integers t, k, and v such that 0 ≤ t ≤ k ≤ v, let Wt 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 Wover(t, -) k (v), which is row-equivalent to Wt k (v). Its Smith normal form is determined. As applications, Wilson's diagonal form of Wt 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 Wt k x = b due to Wilson. Finally we present another inclusion matrix with similar properties to those of Wover(t, -) k (v) which is in some way equivalent to Wt k (v). © 2007 Elsevier Inc. All rights reserved
  6. Keywords:
  7. Smith normal form ; Chains ; Inclusion matrices
  8. Source: Journal of Combinatorial Theory. Series A ; Volume 115, Issue 5 , 2008 , Pages 878-887 ; 00973165 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0097316507001203