Geodesic
Down-step statistics in generalized Dyck paths
Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 1.

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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
Classification : 05A15, 94B10 
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Asinowski, Andrei; Hackl, Benjamin; Selkirk, Sarah J. Down-step statistics in generalized Dyck paths. Discrete mathematics & theoretical computer science, Tome 24 (2022) no. 1. doi : 10.46298/dmtcs.7163. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.7163/

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