Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: SIGNED WORDS AND PERMUTATIONS, II; THE EULER-MAHONIAN POLYNOMIALS. | SIGNED WORDS AND PERMUTATIONS II THE EULER-MAHONIAN POLYNOMIALS Dominique Foata Institut Lothaire 1 rue Murner F-67000 Strasbourg France foata@ Guo-Niu Han . UMR 7501 Université Louis Pasteur et CNRS 7 rue Rene-Descartes F-67084 Strasbourg France guoniu@ Submitted May 6 2005 Accepted Oct 28 2005 Published Nov 7 2005 Mathematics Subject Classifications 05A15 05A30 05E15 Dans la théorie de Morse quand on veut étudier un espace on introduit une fonction numerique puis on aplatit cet espace sur l axe de la valeur de cette fonction. Dans cette operation d aplatissement on cree des singularites de la fonction et celles-ci sont en quelque sorte les vestiges de la topologie qu on a tuée. Rene Thom Logos et Theorie des catastrophes 1982. Dedicated to Richard Stanley on the occasion of his sixtieth birthday. Abstract As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored using the flag-major index on the one hand and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout. 1. Introduction The elements of the hyperoctahedral group Bn n 0 usually called signed permutations may be viewed as words w xix2 . .xn where the letters xi are positive or negative integers and where xiI jx21 . xn is a permutation of 1 2 . n see Bo68 p. 252-253 . For typographical reasons we shall use the notation i i in the sequel. Using the Ỵ-notation that maps each statement A onto the value x A 1 or 0 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 R22 1 depending on whether A is true or not we recall that the usual inversion number inv w of the signed permutation w x1 x2 . . .xn is dehned by invw 2 T X xi xj . 1 j ni j It also makes sense to introduce invw
