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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 conjectureson 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 theother states that $\{W_{n,k}(x)\}_{1\leq k\leq n}$ is a Sturm-unimodal sequencefor any fixed $n\geq 1$. In this paper, we obtain certain recurrence relationsfor $W_{n,k}(x)$, and further confirm their conjectures.

Interlacing property of a family of generating polynomials over Dyck paths