The sorting index and inversion number on order ideals of permutation groups

Björner and Wachs showed that the major index and the inversion number are equidistributed on an order ideal UU of the symmetric group in the weak order if and only if the maximal elements of UU are 132-avoiding permutations. In this paper, we show that the sorting index and the inversion number are equidistributed on an order ideal UU of the symmetric group in the Bruhat order if the maximal elements of UU are 312-avoiding permutations. We also consider the case of type BB. By introducing the notion of BB-increasing subexcedent sequence, we show that the sorting index and inversion number are equidistributed on an order ideal UU of the group of signed permutations in the Bruhat order if the A-code of each maximal element of UU is BB-increasing.