The Sorting Index and Permutation Codes

TL;DR

This paper explores the sorting index in permutation groups, establishes bijections with known permutation codes, and proves equidistribution of various permutation statistics across types A, B, and D.

Contribution

It provides a combinatorial proof connecting the sorting index with Foata-Han codes and extends these results to signed permutations and Coxeter groups of types B and D.

Findings

The bijection of Foata and Han maps (inv, rl-min) to (sor, cyc).

Distribution of (inv_B, Lmap_B, Rmil_B) and (sor_B, Lmap_B, Cyc_B) over signed permutations.

Equidistribution of six pairs of set-valued statistics over signed permutations.

Abstract

In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called the sorting index. Petersen proved that the pairs of statistics $(sor,cyc)$ and $(inv,rl\textrm{-}min)$ have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, and we show that the bijection of Foata and Han serves the purpose of mapping $(inv,rl\textrm{-}min)$ to $(sor,cyc)$. We also give a type $B$ analogue of the Foata-Han bijection, and we derive the quidistribution of $(inv_B,{\rm Lmap_B},{\rm Rmil_B})$ and $(sor_B,{\rm Lmap_B},{\rm Cyc_B})$ over signed permutations. So we get a combinatorial interpretation of Petersen's equidistribution of $(inv_B,nmin_B)$ and $(sor_B,l_B')$. Moreover, we show that the six pairs of set-valued statistics $\rm (Cyc_B,Rmil_B)$, $\rm(Cyc_B,Lmap_B)$, $\rm(Rmil_B,Lmap_B)$, $\rm(Lmap_B,Rmil_B)$, $\rm(Lmap_B,Cyc_B)$ and $\rm(Rmil_B,Cyc_B)$ are equidistributed over signed permutations. For Coxeter groups of type $D$, Petersen showed that the two statistics $inv_D$ and $sor_D$ are equidistributed. We introduce two statistics $nmin_D$ and $\tilde{l}_D'$ for elements of $D_n$ and we prove that the two pairs of statistics $(inv_D,nmin_D)$ and $(sor_D,\tilde{l}_D')$ are equidistributed.