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 [P.A. MacMahon, Combinatory Analysis, vol. 1, Cambridge Univ. Press, 1915], then by Foata by means of a combinatorial bijection [D. Foata, On the Netto inversion number of a sequence, Proc. Amer. Math. Soc. 19 (1968) 236–240]. 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 and Schützenberger [D. Foata, M.-P. Schützenberger, Major index and inversion number of permutations, Math. Nachr. 83 (1978) 143–159], Skandera [Mark Skandera, An Eulerian partner for inversions, Sém. Lothar. Combin. 46 (2001), Article B46d, 19 pages. http://www.mat.univie.ac.at/~slc ], Foata and Han [D. Foata, G.-N. Han, Une nouvelle transformation pour les statistiques Euler–Mahoniennes ensemblistes, Moscow Math. J. 4 (2004) 131–152] and more recently by Hivert, Novelli and Thibon [F. Hivert, J.-C. Novelli, J.-Y. Thibon, Multivariate generalizations of the Foata–Schützenberger equidistribution, 2006, 17 pages. Preprint on arXiv]. 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.