Interlacing Property of a Family of Generating Polynomials over Dyck Paths

Abstract

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.