Proof of a Conjecture on Hankel Determinants for Dyck Paths with Restricted Peak Heights

Abstract

For any integer m ≥ 2 and r ∈ {1, . . . , m}, let f m,r n denote the number of n-Dyck paths whose peak's heights are im + r for some integer i. We find the generating function of f m,r n satisfies a simple algebraic functional equation of degree 2. The r = m case is particularly nice and we give a combinatorial proof. By using the Sulanke and Xin's continued fraction method, we calculate the Hankel determinants for f m,r n . The special case r = m of our result solves a conjecture proposed by Chien, Eu and Fu. We also enriched the class of eventually periodic Hankel determinant sequences.