Year: 2022
Type: article
Abstract: The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.
DOI: 10.46298/dmtcs.7163
Institutions: University of Klagenfurt, Uppsala University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Austrian Science Fund
OpenAlex ID: W3045813272