SIGNED WORDS AND PERMUTATIONS IV

Dominique Foata

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2006/08/10 SIGNED WORDS AND PERMUTATIONS, IV; FIXED AND PIXED POINTS Dominique Foata and Guo-Niu Han Von Jacobs hat er die Statur, Des Rechnens ernstes Fuhren, Von Lottarchen die Frohnatur und Lust zu diskretieren. To Volker Strehl, a dedication a la Goethe, on the occasion of his sixtieth birthday. Abstract The flag-major index “fmaj” and the classical length function “” are used to construct two q-analogs of the generating polynomial for the hyperoctahedral group Bn by number of positive and negative fixed points (resp. pixed points). Specializations of those q-analogs are also derived dealing with signed derange- ments and desarrangements, as well as several classical results that were previ- ously proved for the symmetric group. 1. Introduction The statistical study of the hyperoctahedral group Bn, initiated by Reiner ([Re93a], [Re93b], [Re93c], [Re95a], [Re95b]), has been rejuvenated by Adin and Roichman [AR01] with their introduction of the flag-major index, which was shown [ABR01] to be equidistributed with the length function. See also their recent papers on the subject [ABR05], [ReRo05]. It then appeared natural to extend the numerous results obtained for the symmetric group Sn to the groug Bn. Furthermore, flag-major index and length function become the true q-analog makers needed for calculating various multivariable distributions on Bn.

• signed permutation


• dbn


• let bn


• classical results


• see also


• permutations become plain


• function become


• bn onto

Sujets

• Jacobs
• See also

Informations

• Publié par: pefav
• Nombre de lectures: 16
• Langue: English

Extrait

2006/08/10

SIGNED WORDS AND PERMUTATIONS, IV; FIXED AND PIXED POINTS

Dominique Foata and Guo-Niu Han

Von Jacobs hat er die Statur, Des Rechnens ernstes Fu¨hren, VonLott¨archendieFrohnatur und Lust zu diskretieren. ToVolkerStrehl,adedicationa`laGoethe, on the occasion of his sixtieth birthday.

Abstract The flag-major index “fmaj” and the classical length function “ℓ” are used to construct twoq-analogs of the generating polynomial for the hyperoctahedral groupB n positive and negative fixed points (resp. pixed points).by number of Specializations of thoseq-analogs are also derived dealing with signed derange- ments and desarrangements, as well as several classical results that were previ- ously proved for the symmetric group.

1. Introduction

The statistical study of the hyperoctahedral groupB n , initiated by Reiner ([Re93a], [Re93b], [Re93c], [Re95a], [Re95b]), has been rejuvenated by Adin and Roichman [AR01] with their introduction of theflag-major index, which was shown [ABR01] to be equidistributed with thelength function. See also their recent papers on the subject [ABR05], [ReRo05]. It then appeared natural to extend the numerous results obtained for the symmetric groupS n to the grougB n . Furthermore, flag-major index and length function become the trueq-analog makers needed for calculating various multivariable distributions onB n . In the present paper we start with a generating polynomial forB n by a three-variable statistic involving the number of fixed points (see formula (1.3)) and show that there are two ways ofq-analogizing it, by using the flag-major index on the one hand, and the length function on the other hand. As will be indicated, the introduction of an extra variableZmakes it possible to specialize all our results to the symmetric group. Let us first give the necessary notations. LetB n be the hyperoctahedral group of allsigned permutationsof ordern. The elements ofB n may be viewed as wordsw=x 1 x 2   x n ,

2000Mathematics Subject Classification.Primary 05A15, 05A30, 05E15. Key words and phrases.Hyperoctahedral group, length function, flag-major index, signed permutations, fixed points, pixed points, derangements, desarrangements, pixed factorization.

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DOMINIQUE FOATA AND GUO-NIU HAN

where eachx i belongs to{−n    −11     n}and|x 1 ||x 2 |    |x n |is a permutation of 12   n. Theset(resp. thenumber) ofnegativeletters among thex i ’s is denoted by Negw(resp. negw). Apositive fixed point of the signed permutationw=x 1 x 2   x n is a (positive) integerisuch thatx i =i. It is convenient to writei:=−ifor each integeri. Also, whenAis a set of integers, letA:=i:i∈A}. Ifx i =iwithipositive, we say thatiis anegative fixed pointofw. The set of all positive (resp. negative) fixed points ofwis denoted by Fix + w(resp. Fix − w). Notice that Fix − w⊂Negw. Also let (11) fix + w Fix:= # + w; fix − w:= # Fix − w There are 2 n n! signed permutations of ordern. The symmetric groupS n may be considered as the subset of allwfromB n such that Negw=∅. The purpose of this paper is to providetwoq-analogsfor the polyno- mialsB n (Y 0  Y 1  Z) defined by the identity (12) X u n n ! B n (Y 0  Y 1 n≥0  Z) =  1−u(1 +Z)  −1 × ep x x pe(u ( ( u Y ( 0 + 1+ Y Z 1 ) Z ) )) 

WhenZ= 0, the right-hand side becomes (1−u) −1 exp(uY 0 )exp(u), which is the exponential generating function for the generating polyno- mials for the groupsS n by number of fixed points (see [Ri58], chap. 4). Also, by identification,B n (111) = 2 n n! and it is easy to show (see The- orem 1.1) thatB n (Y 0  Y 1  Z) is in fact the generating polynomial for the groupB n by the three-variable statistic (fix + fix − neg), that is, (13)B n (Y 0  Y 1  Z) = X Y 0 fix + w Y x 1 fi − w Z negw  w∈B n

Recall the traditional notations for theq-ascending factorials (1.4) (a;q) n :=  1( 1 −a)(1−aq)  (1−aq n−1 ) if if n n≥1; =0; (a;q) ∞ := Y (1−aq n−1 ); n≥1 for theq-multinomial coefficients = (q;q) n (m 1 +  +m k =n); (1.5)  m 1 m n k  q ( : q;q) m 1   (q;q) m k and for the twoq-exponentials (see [GaRa90, chap. 1]) (1.6)e q (u) = n X ≥0 (q; u q n ) n ( = u 1 ;q) ∞ ;E q (u) = n X q ( (q 2 n ) ; u q) n = (−u;q) ∞ ≥0 n

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