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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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Advances in Applied Mathematics16, (),297–305
THEkEXTENSION OF A MAHONIAN STATISTIC
BY
∗ GuoNiu)HAN (
ABSTRACT. — Clarke and Foata have recently studied thekextension of several Mahonian statistics. There is an alternate definition for thek Denert statistic that is derived in the present paper.
´ ´ RESUMEece´nemmlC,tekra—R.e´aletFoataont´etudikextension de plusieurs statistiques mahoniennes. Dans cet article, on introduit une autrede´finitionpourlakstatistique de Denert.
Let
1. Introduction
1, (a;q) = n n−1 (1−a)(1−aq). . .(1−aq),
ifn;= 0 ifn≥1 ;
denote theqascending factorialand for each sequencec= (c1, c2, . . . , cr) of nonnegative integers, of summ, let m(q;q)m = c1, c2, . . . , cr(q;q)c1(q;q)c2. . .(q;q)cr
denote theqmultinomial coefficient. Also denote byR(c) the class of allm!/(c1!c2!. . . cr!) rearrangements of the (nondecreasing) word c1c2cr 1 2. . . r. ByMahonian statisticit is meant an integervalued map ping “stat” defined on each classR(c) that satisfies the identity X m statw =q(w∈R(c)). c1, c2, . . . , cr 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, 199495.
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