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 this question, 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 bijection of Foata and Han, and 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 and for elements of D_n and we prove that the two pairs of statistics (inv_D, nmin_D) and (sor_D, ) are equidistributed.

  AMS Classification:  05A05, 05A15, 20F55

  Keywords:  permutation statistic, Mahonian statistic, Coxeter group, set-valued statistic, bijection

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