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

TL;DR

This paper derives algebraic generating functions and Hankel determinants for a class of Dyck paths with restricted peak heights, confirming a conjecture and expanding understanding of Hankel determinant sequences.

Contribution

It provides a new algebraic functional equation for generating functions of Dyck paths with restricted peak heights and proves a conjecture on Hankel determinants for the case r=m.

Findings

Derived algebraic generating functions for f_n^{m,r}

Calculated Hankel determinants using continued fraction methods

Confirmed the conjecture for the case r=m

Abstract

For any integer $m\geq 2$ and $r \in \{1,\dots, m\}$, let $f_n^{m,r}$ denote the number of $n$-Dyck paths whose peak's heights are $im+r$ for some integer $i$. We find the generating function of $f_n^{m,r}$ 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 $f_n^{m,r}$. 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.