Raised $k$-Dyck paths

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 $(0,α)$ to $(\ell,β)$ for all height pairs $α,β\geq 0$, 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 $α< k$ reveal connections with preexisting results concerning height-bounded Dyck paths and "Dyck paths with a negative boundary", respectively.

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