Interlacing Property of a Family of Generating Polynomials over Dyck Paths

In the study of a tantalizing symmetry on Catalan objects, Bóna et al. introduced a family of polynomials \W₍, ₊ (x) \₍ ₊ ₀ defined byW₍, ₊ (x) =₌=₀^kw₍, ₊, ₌x^m, where w₍, ₊, ₌ 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₍, ₊ (x) \₍ ₊ is a Sturm sequence for any fixed k 1, and the other states that \W₍, ₊ (x) \₁ ₊ ₍ is a Sturm-unimodal sequence for any fixed n 1. In this paper, we obtain certain recurrence relations for W₍, ₊ (x), and further confirm their conjectures.