Downstep statistics in generalized Dyck paths
Abstract
The number of downsteps between pairs of upsteps 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.15562
 arXiv:
 arXiv:2007.15562
 Bibcode:
 2020arXiv200715562A
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05A15 (Primary) 05A19;
 05A05 (Secondary)