Enumeration of consecutive patterns in flattened Catalan words

TL;DR

This paper studies the distribution of consecutive patterns in flattened Catalan words, providing explicit formulas for their generating functions, pattern occurrences, and avoidance counts, revealing equivalences and combinatorial insights.

Contribution

It introduces explicit formulas for the joint distribution of multiple patterns in flattened Catalan words, including generating functions and combinatorial equivalences.

Findings

Explicit formulas for generating functions of pattern distributions.

Equivalences in distribution of certain pattern pairs.

Exact counts of pattern occurrences and avoiders.

Abstract

A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has cardinality $\frac{3^{n-1}+1}{2}$ for all $n \geq 1$. In this paper, we consider the distribution of several consecutive patterns on $\mathcal{F}_n$. Indeed, we find explicit formulas for the generating functions of the joint distribution on $\mathcal{F}_n$ of several trios of patterns, along with an auxiliary parameter. As special cases of these formulas, we obtain the generating function for the distribution of all consecutive patterns of length two or three. The following equivalences with regard to being identically distributed on $\mathcal{F}_n$ arise when comparing the various generating functions and may be explained bijectively: $112\approx122$ and $211\approx221\approx231$. In addition, explicit expressions are found for the total number of occurrences on $\mathcal{F}_n$ of each pattern of length two or three as well as for the number of avoiders of each pattern. These results can be obtained as special cases of our more general formulas for the generating functions, but may be explained combinatorially as well, the arguments of which are featured herein.