The sorting index and permutation codes

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-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-min) to (sor, cyc). We also give a type B analogue of the Foata-Han bijection, and we derive the equidistribution of (inv B , Lmap B , Rmil B ) and (sor B , Lmap B , 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 (Cyc B , Rmil B ), (Cyc B , Lmap B ), (Rmil B , Lmap B ), (Lmap B , Rmil B ), (Lmap B , Cyc B ) and (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 andl ′ D for elements of D n and we prove that the two pairs of statistics (inv D , nmin D ) and (sor D ,l ′ D ) are equidistributed.

The sorting index on colored permutations and even-signed permutations

“…By defining the set-valued statistics Poznanović [7] proved that (inv, Rmil, Lmal, Lmap) and (sor, Cyc, Lmal, Lmap) have the same joint distribution over S n,f by means of Foata-Han's bijection in [4]. Analogous results on Coxeter groups of type B and D were also obtained in [7], generalizing the works of Petersen [6] and Chen-Guo-Gong [3].…”

“…For example, let π = (σ, z) = (361475928, (3, 0, 3, 2, 2, 1, 0, 3, 1)) ∈ G 4,9 . Then Lmap(σ) = {1, 2, 5, 7}, Lmal(σ) = {3, 6, 7, 9} and f (π) = (3,6,6,6,7,7,9,9,9). Lmal t (φ(π)) = Lmal(σ ).…”

“…The key ingredient in this section is the bijection ψ : D n → D n , introduced by Chen-Guo-Gone [3], which is the composition…”

“…It is clear that f defines a Ferrers shape and S n,f consists of those permutations corresponding to non-attacking rook placements. For example, let f = (2,3,3,4), then S 4,f = {1234, 1324, 2134, 2314}, as illustrated in Figure 1. By defining the set-valued statistics Poznanović [7] proved that (inv, Rmil, Lmal, Lmap) and (sor, Cyc, Lmal, Lmap) have the same joint distribution over S n,f by means of Foata-Han's bijection in [4].…”

“…where h i is the smallest possible index at which the letter i can appear in a colored permutation σ ∈ G r,n,f . In the above example f = (2,3,3,4) and hence H(f ) = (1,1,2,4). The superindex of a statistic is taken as mod r, for example, Rmil −2 (σ) = Rmil 3 (σ) if r = 5.…”