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

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 $(a_1,a_2,\dots ,a_k)$ 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 $k$, $1\le i\le k$. The proofs given are done via bijective means or with generating functions.

References in corpus (2)

• Refined enumeration of $k$-plane trees and $k$-noncrossing trees
• Down-step statistics in generalized Dyck paths

Cited by in corpus (1)

• Refined Lattice Path Enumeration and Combinatorial Reciprocity