A combinatorial proof of the parity unimodality of the [formula omitted]-rational [formula omitted]-Catalan polynomial for [formula omitted

Abstract

A sequence a0,a1,a2,⋯ is parity unimodal if it exists k1 and k2 such that a0≤a2≤⋯≤ak1≥ak1+2≥⋯ and a1≤a3≤⋯≤ak2≥ak2+2≥⋯. A polynomial f(q)=∑i=0naiqi is parity unimodal if its coefficient sequence a0,a1,⋯,an is parity unimodal. Recently, Xin and Zhong conjectured that the (m,n)-rational q-Catalan polynomial is parity unimodal. They showed that this conjecture holds for m≤5 using the constant term method and generating functions. In this paper, we give a combinatorial proof of the m=3 case.