A Combinatorial Proof of the Unimodality and Symmetry of Weak Composition Rank Sequences

A weak composition of an integer s with m parts is a way of writing s as the sum of a sequence of non-negative integers of length m. Given two positive integers m and n, let N(m, n) denote the set of all weak compositions \(\alpha =(\alpha _1,\dots ,\alpha _m)\) with \(0 \le \alpha _i \le n\) for \(1 \le i \le m\) and \(c_w^{m,n}(s)\) be the number of weak composition of s into m parts with no part exceeding n. A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. In this paper, we show that the poset N(m, n) can be expressed as a disjoint of symmetric chains by constructive method, which implies that its rank sequence \(c_w^{m,n}(0),c_w^{m,n}(1),\dots ,c_w^{m,n}(mn)\) is unimodal and symmetric.