Euler-Mahonian triple set-valued statistics on permutations

Abstract

The inversion number and the major index are equidistributed on the symmetric group. This is a classical result, first proved by MacMahon, then by Foata by means of a combinatorial bijection. Ever since many refinements have been derived, which consist of adding new statistics, or replacing integral-valued statistics by set-valued ones. See the works by Foata-Schutzenberger, Skandera, Foata-Han and more recently by Hivert-Novelli-Thibon. In the present paper we derive a general equidistribution property on Euler-Mahonian set-valued statistics on permutations, which unifies the above four refinements. We also state and prove the so-called "complement property" of the Majcode.