Pattern expansions of permutation statistics

TL;DR

This paper investigates how permutation statistics can be expanded in terms of pattern occurrence functions, establishing finiteness and positivity criteria, and providing combinatorial interpretations for specific cases.

Contribution

It generalizes previous results by proving finiteness of pattern expansions for a broad class of permutation statistics and introduces a criterion for their positivity.

Findings

Finiteness of pattern expansions for higher moment permutation statistics

A combinatorial criterion for positivity of pattern expansions

Enumeration interpretation for coefficients of reduced words

Abstract

We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the higher moment statistics, generalizing a result of Berman and Tenner. We also give a combinatorial criterion for the positivity of pattern expansions. Using this criterion, we show that the pattern expansion of the number of reduced words of a permutation is positive and give an enumerative interpretation for the coefficients.