2022/06/02 by Paul Drube, Drube, Paul
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2206.01194
openalex publication_date 2022/06/02 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
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 (ℓ,β) for all height pairs α,β≥ 0, all lengths ℓ ≥ 0, and all k ≥ 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 Ck(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.