On Parity Unimodality of $q$-Catalan Polynomials

Abstract

A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\leqslant a_1\leqslant \cdots \leqslant a_k\geqslant a_{k+1} \geqslant \cdots \geqslant a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= \frac{1}{[n+m]} {m+n \brack n}_q $ 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 our conjecture for $m\le 5$ in a straightforward way.