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arXiv · 2007.15562

Down-step statistics in generalized Dyck paths

Andrei Asinowski·Benjamin Hackl·Sarah J. Selkirk
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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.

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Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk. 2020-07-30. Down-step statistics in generalized Dyck paths. https://doi.org/10.46298/dmtcs.7163

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