Descent distribution on Catalan words avoiding a pattern of length at most three

TL;DR

This paper analyzes the distribution of descents in Catalan words that avoid certain patterns of length up to three, providing generating functions and enumerations, some of which are novel contributions to combinatorics.

Contribution

It introduces explicit bivariate generating functions for descents in pattern-avoiding Catalan words, including new enumerations not previously recorded.

Findings

Derived generating functions for each pattern p

Enumerated pattern-avoiding Catalan words

Identified new sequences not in OEIS

Abstract

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set. Some of the obtained enumerating sequences are not yet recorded in the On-line Encyclopedia of Integer Sequences.