Pattern expansions of permutation statistics

TL;DR

The paper develops a unified framework for expressing permutation statistics as pattern-count expansions and analyzes when these expansions are finite or nonnegative. It proves a general averaging criterion via marked permutations that yields pattern-finiteness for broad classes of statistics, including higher moment statistics with coefficients tied to Stirling numbers. It then introduces an inclusion-exclusion approach to obtain pattern-positivity, with the number of reduced words $\\mathsf{rw}(w)$ as a central case; this yields an explicit, combinatorially interpretable expansion and a decomposition of $RW(w)$ into pattern-indexed contributions. Together, these results offer new tools for bounding and interpreting permutation statistics through pattern-level data and suggest rich avenues for connecting pattern expansions to classical algebraic combinatorics questions such as Schubert positivity.

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.

Figures (3)

• Figure 1: The reduced word $s_2 s_1 s_2 s_3 s_2\in \mathsf{RW}(3421)$ restricts to $s_1s_2\in \mathsf{RW}(231)$ for the $231$-pattern that occurs on indices $\{1,2,4\}$ in $3421$.
• Figure 2: The minimal lift of the reduced word $s_2s_1s_2\in \mathsf{RW}(321)$ from the $321$-pattern that occurs on indices $\{1,3,4\}$ in $4231$ is $s_3 s_1 s_2 s_3 s_1\in \mathsf{RW}(4231)$.
• Figure 3: The five wiring diagrams for the permutation $3421$ with dashed wires corresponding to nonessential indices, corresponding to the expansion of $\mathsf{rw}(3421)$ from the pattern expansion of $\mathsf{rw}$.

Theorems & Definitions (47)

• Example 1.1: BermanTenner
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: cf. BrandenClaesson
• proof : Proof of Proposition \ref{['prop:every statistic has a pattern expansion']}
• Definition 2.6
• Example 2.7
• Lemma 2.8