On Parity Unimodality of $q$-Catalan Polynomials
On Parity Unimodality of $q$-Catalan Polynomials
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 …