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arXiv · 2206.01194

Raised $k$-Dyck paths

Paul Drube
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Abstract

Raised $k$-Dyck paths are a generalization of $k$-Dyck paths that may both begin and end at a nonzero height. In this paper, we develop closed formulas for the number of raised $k$-Dyck paths from $(0,\alpha)$ to $(\ell,\beta)$ for all height pairs $\alpha,\beta \geq 0$, all lengths $\ell \geq 0$, and all $k \geq 2$. We then enumerate raised $k$-Dyck paths with a fixed number of returns to ground, a fixed minimum height, and a fixed maximum height, presenting generating functions (in terms of the generating functions $C_k(t)$ for the $k$-Catalan numbers) when closed formulas aren't tractable. Specializing our results to $k=2$ or to $\alpha < k$ reveal connections with preexisting results concerning height-bounded Dyck paths and "Dyck paths with a negative boundary", respectively.

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Paul Drube. 2022-06-02. Raised $k$-Dyck paths. https://arxiv.org/abs/2206.01194

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