Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$

TL;DR

This paper provides an exact enumeration formula for generalized Dyck paths, called $k_t$-Dyck paths, based on the distribution of down-steps at heights modulo $k$, using bijective and generating function methods.

Contribution

It introduces a new enumeration formula for $k_t$-Dyck paths considering down-steps at heights modulo $k$, expanding combinatorial understanding of these paths.

Findings

Derived an exact enumeration formula for $k_t$-Dyck paths

Provided bijective and generating function proofs

Enhanced understanding of path distributions at specific heights

Abstract

For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-steps at a height of $i$ modulo $k$. Given a tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration formula for the number of $k_t$-Dyck paths of length $(k+1)n$ with $a_i$ down-steps at a height of $i$ modulo $k$, $1 \leq i \leq k$. The proofs given are done via bijective means or with generating functions.