Raised k-Dyck Paths
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,
(Concerned with sequences
Received May 24 2023;
revised version received June 6 2023.
Published in Journal of Integer Sequences,
June 10 2023.
Journal of Integer Sequences home page