Mesh patterns and the expansion of permutation statistics as sums of permutation patterns

TL;DR

This paper introduces mesh patterns to expand permutation statistics as sums of permutation patterns, providing explicit formulas and revealing natural occurrences of special permutations and Mahonian statistics.

Contribution

It defines mesh patterns and derives explicit expansion formulas for permutation statistics, connecting them to pattern avoidance and introducing new Mahonian statistics.

Findings

Explicit expansion formulas for permutation statistics using mesh patterns

Identification of natural occurrences of special permutations as pattern avoiders

Introduction of new Mahonian permutation statistics

Abstract

Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}p^{\star}(\tau)$ where $p^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, Andr\'e permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.