Fix Mahonian Calculus II: further statistics

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2007/08/30 Fix-Mahonian Calculus, II: further statistics Dominique Foata and Guo-Niu Han ABSTRACT. Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (fix,des,maj), or the pair (fix,maj), where “fix,” “des” and “maj” denote the number of fixed points, the number of descents and the major index, respectively. 1. Introduction First, recall the traditional notations for the q-ascending factorials (a; q)n := { 1, if n = 0; (1? a)(1? aq) · · · (1? aqn?1), if n ≥ 1; (a; q)∞ := ∏ n≥1 (1? aqn?1); and the q-exponential (see [GaRa90, chap. 1]) eq(u) = ∑ n≥0 un (q; q)n = 1(u; q)∞ . Furthermore, let (An(Y, t, q)) and (An(Y, q)) (n ≥ 0) be the sequences of polynomials respectively defined by the factorial generating functions ∑ n≥0 An(Y, t, q) un (t; q)n+1 := ∑ s≥0 ts ( 1? u s ∑ i=

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Publié par

pefav

Nombre de lectures

Langue

English

2007/08/30

Fix-Mahonian Calculus, II: further statistics Dominique Foata and Guo-Niu Han A BSTRACT . Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (ﬁx,des,maj), or the pair (ﬁx,maj), where “ﬁx,” “des” and “maj” denote the number of ﬁxed points, the number of descents and the major index, respectively.

1. Introduction First, recall the traditional notations for the q -ascending factorials ( a ; q ) n :=  (11  − a )(1 − aq )    (1 − aq n − 1 )  iiff nn ≥ =10;; ( a ; q ) ∞ := Y (1 − aq n − 1 ); n ≥ 1 and the q -exponential (see [GaRa90, chap. 1])  e q ( u ) = n X ≥ 0 ( q ; uq n ) n =( u ;1 q ) ∞ Furthermore, let ( A n ( Y t q )) and ( A n ( Y q )) ( n ≥ 0) be the sequences of polynomials respectively deﬁned by the factorial generating functions (1  1) n X ≥ 0 A n ( Y t q )( t ; qu ) nn +1 := s ≥ X 0 t s  1 − u i = s X 0 q i  − 1 (( uuY ;; qq )) ss ++11 ; (1  2) n X ≥ 0 A n ( Y q )( q ; uq n ) n :=  1 − 1 u − q  − 1 (( uuY ;; qq )) ∞∞  Of course, (1.2) can be derived from (1.1) by letting the variable t tend to 1, so that A n ( Y q ) = A n ( Y 1  q ). The classical combinatorial interpretation for those classes of polynomials has been found by Gessel and Reutenauer [GeRe93] (see Theorem 1.1 below). For each permutation σ = σ (1) σ (2)    σ ( n ) from the symmetric group S n let i σ := σ − 1 denote 1

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