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arXiv · 2204.14023

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

Clemens Heuberger·Sarah J. Selkirk·Stephan Wagner
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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.

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Clemens Heuberger, Sarah J. Selkirk, Stephan Wagner. 2022-04-29. Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$. https://doi.org/10.37236/11218

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