Journal of Integer Sequences, Vol. 26 (2023), Article 23.6.7

Raised k-Dyck Paths

Paul Drube
Valparaiso University
Valparaiso, IN 46383


Raised k-Dyck paths are a generalization of k-Dyck paths that may both begin and end at 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. This represents a new approach to the enumeration of "simple paths with linear boundaries of rational slope", as discussed by Krattenthaler in his Handbook of Enumerative Combinatorics. We then expand upon Krattenthaler's results by enumerating raised k-Dyck paths with a fixed number of returns to ground, a fixed minimum height, and a fixed maximum height, presenting generating functions when closed formulas are not tractable. Specializing our results to either k = 2 or to α < k reveal further connections with preexisting results about height-bounded Dyck paths and "Dyck paths with a negative boundary", respectively.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000108 A000245 A000340 A000588 A001519 A001764 A001835 A002057 A002293 A003517 A003518 A004253 A006013 A006629 A006630 A006632 A006633 A024175 A026012 A026013 A026014 A026016 A026017 A026018 A026026 A026027 A026029 A026030 A026031 A030983 A033191 A069271 A080937 A080938 A081704 A083881 A102893 A124302 A143648 A196678 A211216 A261399 A334608 A334680 A334682 A334976 A334977 A336945.)

Received May 24 2023; revised version received June 6 2023. Published in Journal of Integer Sequences, June 10 2023.

Return to Journal of Integer Sequences home page