On Parity Unimodality of $q$-Catalan Polynomials

Abstract

A polynomial $A(q)={\sum }_{i=0}^n{a}_i{q}^i$ is said to be unimodal if $a_0\le a_1\le \cdots \le a_k\ge a_{k+1}\ge \cdots \ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)=\frac{1}{[n+m]}\left[\genfrac{}{}{0ex}{}{m+n}{n}\right]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $m\le 5$ in a straightforward way.

16 pages, 3 figures