Inclusion Matrices and Chains
| dc.creator | Ghorbani, E. | |
| dc.creator | Khosrovshahi, G. B. | |
| dc.creator | Maysoori, Ch. | |
| dc.creator | Mohammad-Noori, M. | |
| dc.date | 2007-09-20 | |
| dc.date.accessioned | 2026-07-07T08:31:01Z | |
| dc.date.available | 2026-07-07T08:31:01Z | |
| dc.description | 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 $W_{tk}(v)$. | |
| dc.description | Accepted for publication in Journal of Combinatorial Theory, Series A | |
| dc.identifier | https://arxiv.org/abs/0709.3144 | |
| dc.identifier | http://arxiv.org/abs/0709.3144 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138393 | |
| dc.subject | Combinatorics | |
| dc.subject | 05B05, 05B20, 15A21, 05D05 | |
| dc.title | Inclusion Matrices and Chains | |
| dc.type | text |