Cycles and sorting index for matchings and restricted permutations

TL;DR

This paper establishes new combinatorial equidistribution results for permutation statistics related to matchings and restricted permutations, introducing a sorting index and using bijections with Dyck paths.

Contribution

It introduces a new sorting index for matchings and restricted permutations, providing combinatorial proofs and refined equidistribution results.

Findings

Mahonian-Stirling pairs are equidistributed on rook arrangements

A new sorting index for matchings is defined and used

Refined results describe minimal elements in cycles and right-to-left minima

Abstract

We prove that the Mahonian-Stirling pairs of permutation statistics $(\sor, \cyc)$ and $(\inv, \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sorting index for bicolored matchings and use it to show analogous equidistribution results for restricted permutations of type $B_n$ and $D_n$.