{"docId":9558,"paperId":7163,"url":"https:\/\/dmtcs.episciences.org\/7163","doi":"10.46298\/dmtcs.7163","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":608,"name":"vol. 24, no. 1"}],"section":[{"sid":6,"title":"Combinatorics","description":[]}],"repositoryName":"arXiv","repositoryIdentifier":"2007.15562","repositoryVersion":6,"repositoryLink":"https:\/\/arxiv.org\/abs\/2007.15562v6","dateSubmitted":"2021-02-08 18:21:11","dateAccepted":"2022-04-21 16:11:36","datePublished":"2022-05-24 16:31:23","titles":["Down-step statistics in generalized Dyck paths"],"authors":["Asinowski, Andrei","Hackl, Benjamin","Selkirk, Sarah J."],"abstracts":["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."],"keywords":["Mathematics - Combinatorics","Computer Science - Discrete Mathematics","05A15 (Primary) 05A19, 05A05 (Secondary)"]}