On Parity Unimodality of q-Catalan Polynomials - Xin, Guoce; Zhong, Yueming (The Electronic Journal of Combinatorics, 2020)
A polynomial A(q)=_i=0^n a_iq^i is said to be unimodal if a_0a_1… a_ka_k+1 … a_n. We investigate the unimodality of rational q-Catalan polynomials, which is defined to be C_m,n(q)= 1[n+m] m+n 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 m5 in a straightforward way.