Catalan words avoiding pairs of length three patterns

TL;DR

This paper studies Catalan words that avoid pairs of length-3 patterns, providing structural insights and enumeration results, some of which offer new combinatorial interpretations for known sequences.

Contribution

It systematically analyzes Catalan words avoiding pairs of length-3 patterns, extending previous enumeration work with recursive, bijective, and generating function methods.

Findings

Enumeration of certain pattern-avoiding Catalan words

Structural properties of these words elucidated

New combinatorial interpretations for known sequences

Abstract

Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.