Multivariate generalizations of the Foata-Schutzenberger equidistribution

A result of Foata and Schutzenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property: the sorted vectors of the Lehmer code, of the inverse majcode, and of a new code (the inverse saillance code), have the same distribution on a descent class, and their common multivariate generating function is a flagged ribbon Schur function.