On Hankel Determinants for Dyck Paths with Peaks Avoiding Multiple Classes of Heights

TL;DR

This paper investigates Hankel determinants associated with Dyck paths avoiding certain peak heights, providing explicit formulas and conditions for periodicity based on the structure of the height sets.

Contribution

It offers explicit descriptions of Hankel determinants for Dyck paths with peaks avoiding specific height classes and establishes conditions for their periodicity.

Findings

Explicit formulas for Hankel determinants in terms of arithmetic progressions

Identification of periodic sequences of Hankel determinants for various sets

Sufficient conditions for the periodicity of Hankel determinants based on set structure

Abstract

For any integer $m\ge 2$ and a set $V\subset \{1,\dots,m\}$, let $(m,V)$ denote the union of congruence classes of the elements in $V$ modulo $m$. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set $(m,V)$. For any set $V$ of even elements of an even modulo $m$, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied $(m,V)$ with periodic sequences of Hankel determinants. We present a sufficient condition for the set $(m,V)$ such that the sequence of Hankel determinants is periodic, including even and odd modulus $m$.