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Signed words and permutations, II; The Euler-Mahonian polynomials

By Dominique Foata and Guo-niu Han


Dans la théorie de Morse, quand on veut étudier un espace, on introduit une fonction numérique; puis on aplatit cet espace sur l’axe de la valeur de cette fonction. Dans cette opération d’aplatissement, on crée des singularités de la fonction et celles-ci sont en quelque sorte les vestiges de la topologie qu’on a tuée. René Thom, Logos et Théorie des catastrophes, 1982. Dedicated to Richard Stanley, on the occasion of his sixtieth birthday. 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

Year: 2009
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