  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 $W_{tk}(v)$.
