Down-step statistics in generalized Dyck paths
Combinatorics
2023-06-22 v6 Discrete Mathematics
Abstract
The number of down-steps between pairs of up-steps in -Dyck paths, a generalization of Dyck paths consisting of steps such that the path stays (weakly) above the line , 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 -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
Cite
@article{arxiv.2007.15562,
title = {Down-step statistics in generalized Dyck paths},
author = {Andrei Asinowski and Benjamin Hackl and Sarah J. Selkirk},
journal= {arXiv preprint arXiv:2007.15562},
year = {2023}
}