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.
@article{10_37236_8870,
author = {Guoce Xin and Yueming Zhong},
title = {On parity unimodality of {\(q\)-Catalan} polynomials},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {1},
doi = {10.37236/8870},
zbl = {1431.05011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8870/}
}
TY - JOUR
AU - Guoce Xin
AU - Yueming Zhong
TI - On parity unimodality of \(q\)-Catalan polynomials
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8870/
DO - 10.37236/8870
ID - 10_37236_8870
ER -
%0 Journal Article
%A Guoce Xin
%A Yueming Zhong
%T On parity unimodality of \(q\)-Catalan polynomials
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8870/
%R 10.37236/8870
%F 10_37236_8870
Guoce Xin; Yueming Zhong. On parity unimodality of \(q\)-Catalan polynomials. The electronic journal of combinatorics, Tome 27 (2020) no. 1. doi: 10.37236/8870
