In the study of a tantalizing symmetry on Catalan objects, Bóna et al. introduced a family of polynomials ,k(x)≥ k≥ 0 defined byWn,k(x)=∑ₘ₌₀ᵏwn,k,mxᵐ,where wn,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 ,k(x)≥ k is a Sturm sequence for any fixed k≥ 1, and the other states that ,k(x)\1≤ k≤ n is a Sturm-unimodal sequence for any fixed n≥ 1. In this paper, we obtain certain recurrence relations for Wn,k(x), and further confirm their conjectures.
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Wang et al. (2024) studied this question.