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 $\sum\limits_{i=1}^{m}\beta_i <\sum\limits_{i=1}^{m}\alpha_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.