Advances in Applied Mathematics

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Advances in Applied Mathematics 16, (????),297–305 THE k-EXTENSION OF A MAHONIAN STATISTIC BY Guo-Niu HAN (?) ABSTRACT. — Clarke and Foata have recently studied the k-extension of several Mahonian statistics. There is an alternate definition for the k- Denert statistic that is derived in the present paper. RESUME. — Recemment, Clarke et Foata ont etudie la k-extension de plusieurs statistiques mahoniennes. Dans cet article, on introduit une autre definition pour la k-statistique de Denert. 1. Introduction Let (a; q)n = { 1, if n = 0; (1? a)(1? aq) . . . (1? aqn?1), if n ≥ 1 ; denote the q-ascending factorial and for each sequence c = (c1, c2, . . . , cr) of non-negative integers, of sum m, let [ m c1, c2, . . . , cr ] = (q; q)m(q; q)c1(q; q)c2 . . . (q; q)cr denote the q-multinomial coefficient. Also denote by R(c) the class of all m! /(c1! c2! . .

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• large letters

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• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

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Advances in Applied Mathematics16, (),297–305

THEkEXTENSION OF A MAHONIAN STATISTIC

BY

∗ GuoNiu)HAN (

A BSTRACT . — Clarke and Foata have recently studied thekextension of several Mahonian statistics. There is an alternate definition for thek Denert statistic that is derived in the present paper.

´ ´ R ESUME ece´nemmlC,tekra—R.e´aletFoataont´etudikextension de plusieurs statistiques mahoniennes. Dans cet article, on introduit une autrede´finitionpourlakstatistique de Denert.

Let

1. Introduction

 1, (a;q) = n n −1 (1−a)(1−aq). . .(1−aq),

ifn;= 0 ifn≥1 ;

denote theqascending factorialand for each sequencec= (c 1 , c 2 , . . . , c r ) of nonnegative integers, of summ, let   m(q;q) m = c 1 , c 2 , . . . , c r (q;q) c 1 (q;q) c 2 . . .(q;q) c r

denote theqmultinomial coefficient. Also denote byR(c) the class of allm!/(c 1 !c 2 !. . . c r !) rearrangements of the (nondecreasing) word c 1 c 2 c r 1 2. . . r. ByMahonian statisticit is meant an integervalued map ping “stat” defined on each classR(c) that satisfies the identity   X m statw =q(w∈R(c)). c 1 , c 2 , . . . , c r w

Theinversion number“inv” and themajor index“maj” are classical examples of Mahonian statistics (see [M]). Another example of such a statistic is provided with theDenert statistic, “den”, introduced recently (see [Den]) in the study of hereditary orders in central simple algebras. The statistics “maj” and “den” have two definitions, one “natural” or

∗ ( ) Supported in part by a grant of the European Community Programme on Algebraic Combinatorics, 199495.

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