Fix-Mahonian calculus, I: Two transformations

We construct two bijections of the symmetric group S n onto itself that enable us to show that three new three-variable statistics are equidistributed with classical statistics involving the number of fixed points. The first one is equidistributed with the triplet ( fix , des , maj ) , the last two with ( fix , exc , maj ) , where “fix”, “des”, “exc” and “maj” denote the number of fixed points, the number of descents, the number of excedances and the major index, respectively.