Interlacing property of a family of generating polynomials over Dyck paths

TL;DR

This paper proves conjectures about the interlacing and Sturm properties of a family of polynomials derived from Dyck paths, revealing new structural insights into Catalan objects.

Contribution

It establishes recurrence relations and confirms conjectures on the interlacing and Sturm sequence properties of these polynomials.

Findings

Confirmed that W_{n,k}(x) forms a Sturm sequence for fixed k.

Demonstrated that W_{n,k}(x) is a Sturm-unimodal sequence for fixed n.

Provided recurrence relations for the polynomials.

Abstract

In the study of a tantalizing symmetry on Catalan objects, B\'ona et al. introduced a family of polynomials $\{W_{n,k}(x)\}_{n\geq k\geq 0}$ defined by \begin{align*} W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^{m}, \end{align*} 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.