Fix-mahonian calculus III; a quadruple distribution

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points “fix”, the second the so-called “pix” statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where “des” and “exc” stand for the number of descents and excedances, respectively. Authors’ addresses: Dominique Foata, Institut Lothaire, 1 rue Murner, F-67000 Strasbourg, France; Guo-Niu Han, Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China; I.R.M.A. UMR 7501, Université Louis Pasteur et CNRS, 7, rue René-Descartes, F-67084 Strasbourg, France