Raised $k$-Dyck paths

TL;DR

This paper introduces formulas and enumerations for raised k-Dyck paths, a generalization of Dyck paths that start and end at nonzero heights, connecting to existing path enumeration results.

Contribution

Develops closed formulas and generating functions for raised k-Dyck paths with various constraints, extending classical Dyck path enumeration.

Findings

Closed formulas for path counts between arbitrary heights.

Generating functions for paths with fixed returns and height bounds.

Connections to existing Dyck path enumeration results.

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.