Cycles and sorting index for matchings and restricted permutations

Svetlana Poznanović

doi:10.46298/dmtcs.2338

Abstract

We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed 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.

Highlights

An inversion in a permutation σ is a pair σ(i) > σ(j) such that i < j
All permutation statistics that are distributed with inv are called Mahonian
MacMahon’s remarkable result initiated a systematic research of permutation statistics and in particular many more Mahonian statistics have been described in the literature since

Summary

Introduction

An inversion in a permutation σ is a pair σ(i) > σ(j) such that i < j. Petersen showed that qsor(σ)tcyc(σ) = t(t + q)(t + q + q2) · · · (t + q + · · · + qn−1), σ∈Sn which implies equidistribution of the pairs (inv, rlmin) and (sor, cyc). Petersen defined sorD, a sorting index for type Dn permutations and showed that it is equidistributed with the number of type Dn inversions: n−1. While space constraints prevent us from providing details in this extended abstract, we mention that in [8] we define a sorting index and cycle number for bicolored matchings in a fashion analogous to what we will show for ordinary matchings This gives a combinatorial proof that the pairs (sorB, B) and (invB, nminB) are equidistributed on the set of restricted signed permutations.

Statistics on perfect matchings

Statistics based on crossings and nestings

Cycles and sorting index for matchings

Connections with restricted permutations