Interlacing Property of a Family of Generating Polynomials over Dyck Paths

In the study of a tantalizing symmetry on Catalan objects, B & oacute;na et al. introduced a family of polynomials {W (n,k)(x)}n >= k >= 0 defined by<br /> W-n,W-k(x) = (k)& sum;(m=0 )w(n,k),(m)x(m),<br /> 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,W-k(x)}n >= k is a Sturm sequence for any fixed k >= 1, and the other states that {W-n,W-k(x)}1 <= k <= n is a Sturm-unimodal sequence for any fixed n >= 1. In this paper, we obtain certain recurrence relations for W-n,W-k(x), and further confirm their conjectures.