Popularity of patterns over $d$-equivalence classes of words and permutations

TL;DR

This paper investigates the relationship between pattern popularity and $d$-equivalence classes of words and permutations, revealing that $d$-equivalent patterns are equally popular across these classes, a non-trivial combinatorial property.

Contribution

It proves that $d$-equivalent patterns are equipopular over any $d$-equivalence class, establishing a novel connection between pattern popularity and $d$-equivalence.

Findings

$d$-equivalent patterns are equipopular over any $d$-equivalence class

Equipopularity does not follow trivially from equidistribution

The result links pattern popularity with combinatorial equivalence classes

Abstract

Two same length words are $d$-equivalent if they have same descent set and same underlying alphabet. In particular, two same length permutations are $d$-equivalent if they have same descent set. The popularity of a pattern in a set of words is the overall number of copies of the pattern within the words of the set. We show the far-from-trivial fact that two patterns are $d$-equivalent if and only if they are equipopular over any $d$-equivalence class, and this equipopularity does not follow obviously from a trivial equidistribution.