Down-step statistics in generalized Dyck paths

TL;DR

This paper studies down-step statistics in generalized Dyck paths, revealing new identities, connections to coding theory, and a novel interpretation of Catalan numbers through bijective and generating function methods.

Contribution

It introduces a comprehensive analysis of down-steps in $k_t$-Dyck paths, establishing new identities and linking to coding theory and Catalan numbers.

Findings

Derived identities for down-step statistics in $k_t$-Dyck paths

Established bijective and generating function proofs

Connected $k_t$-Dyck paths to punctured convolutional codes

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.