# Descent distribution on Catalan words avoiding ordered pairs of Relations

## TL;DR

This paper derives generating functions and bijections for counting descents in Catalan words avoiding specific pairs of relations, extending enumerative combinatorics in pattern-avoiding structures.

## Contribution

It systematically provides bivariate generating functions and bijections for Catalan words avoiding pairs of relations, advancing pattern avoidance enumeration.

## Key findings

- Derived bivariate generating functions for descent counts
- Constructed bijections preserving the number of descents
- Calculated total descents on pattern-avoiding Catalan words

## Abstract

This work is a continuation of some recent articles presenting enumerative results for Catalan words avoiding one or a pair of consecutive or classical patterns of length $3$. More precisely, we provide systematically the bivariate generating function for the number of Catalan words avoiding a given pair of relations with respect to the length and the number of descents. We also present several constructive bijections preserving the number of descents. As a byproduct, we deduce the generating function for the total number of descents on all Catalan words of a given length and avoiding a pair of ordered relations.