A symmetric chain decomposition of $N(m,n)$ of composition

Abstract

A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\alpha =({\alpha }_1,\cdots ,{\alpha }_m)$, with $0\le {\alpha }_i\le n$ for each $i=1,\cdots ,m$. Define order $<$ as follow, $\forall \alpha ,\beta \in N(m,n)$, $\beta <\alpha$ if and only if ${\beta }_i\le {\alpha }_i(i=1,\cdots ,m)$ and $\sum _{i=1}^m{\beta }_i<\sum _{i=1}^m{\alpha }_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.

10 pages, 7 figures