Pattern statistics in faro words and permutations

TL;DR

This paper investigates pattern distributions in faro words, establishing bijections with Dyck paths, and deriving generating functions for pattern statistics, including special subclasses like permutations and involutions.

Contribution

It introduces a bijection between faro words and dispersed Dyck paths, enabling new enumerative results for pattern distributions and popularities.

Findings

Derived multivariate generating functions for pattern distributions.

Mapped pattern statistics in faro words to path statistics via bijection.

Analyzed subclasses such as permutations, involutions, and derangements.

Abstract

We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.