In the study of a tantalizing symmetry on Catalan objects, Bóna et al. introduced a family of polynomials $\{W_{n,k}(x)\}_{n\geq k\geq 0}$ defined by$$W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^{m},$$where $w_{n,k,m}$ counts the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. They proposed two conjectures on the interlacing property of these polynomials, one of which states that $\{W_{n,k}(x)\}_{n\geq k}$ is a Sturm sequence for any fixed $k\geq 1$, and the other states that $\{W_{n,k}(x)\}_{1\leq k\leq n}$ is a Sturm-unimodal sequence for any fixed $n\geq 1$. In this paper, we obtain certain recurrence relations for $W_{n,k}(x)$, and further confirm their conjectures.
@article{10_37236_12375,
author = {Bo Wang and Candice X.T. Zhang},
title = {Interlacing property of a family of generating polynomials over {Dyck} paths},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {1},
doi = {10.37236/12375},
zbl = {1536.05024},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12375/}
}
TY - JOUR
AU - Bo Wang
AU - Candice X.T. Zhang
TI - Interlacing property of a family of generating polynomials over Dyck paths
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/12375/
DO - 10.37236/12375
ID - 10_37236_12375
ER -
%0 Journal Article
%A Bo Wang
%A Candice X.T. Zhang
%T Interlacing property of a family of generating polynomials over Dyck paths
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/12375/
%R 10.37236/12375
%F 10_37236_12375
Bo Wang; Candice X.T. Zhang. Interlacing property of a family of generating polynomials over Dyck paths. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/12375
