Cycle type and descent set in wreath products

We express the number of elements of the hyperoctahedral group Bn, which have descent set K and such that their inverses have descent set J, as a scalar product of two representations of Bn. We also give the number of elements of Bn, which have a prescribed descent set and which are in a given conjugacy class of Bn by another scalar product of representations of Bn. For this, we first establish corresponding results for certain wreath products. We finally give, by generating series of symmetric functions, some analogs of the classical formulas which express the exponential generating series of alternating elements in the Bn's.

[1]  Jim Pitman,et al.  Riffle shuffles, cycles, and descents , 1995, Comb..

[2]  C. Reutenauer.  Free Lie Algebras , 1993 .

[3]  Henri Poincaré,et al.  Sur les propriétés arithmétiques des courbes algébriques , 1901 .

[4]  Ladnor Geissinger,et al.  Representations of the hyperoctahedral groups , 1978 .

[5]  Louis Solomon,et al.  A decomposition of the group algebra of a finite Coxeter group , 1968 .

[6]  Ira M. Gessel,et al.  Counting Permutations with Given Cycle Structure and Descent Set , 1993, J. Comb. Theory A.

[7]  I. G. MacDonald,et al.  Polynomial functors and wreath products , 1980 .

[8]  A. Kerber.  Representations of permutation groups , 1975 .

[9]  J. Stembridge.  The projective representations of the hyperoctahedral group , 1992 .

[10]  W. Specht.  Eine Verallgemeinerung der symmetrischen Gruppe , 1932 .

[11]  Désiré André,et al.  Sur les permutations alternées , 1881 .

[12]  Einar Steingrímsson,et al.  Permutation Statistics of Indexed Permutations , 1994, Eur. J. Comb..

[13]  Dominique Foata,et al.  Major Index and Inversion Number of Permutations , 1978 .

[14]  Richard Ehrenborg,et al.  Sheffer posets and r-signed permutations , 1995 .

[15]  H. O. Foulkes,et al.  Enumeration of permutations with prescribed up-down and inversion sequences , 1976, Discret. Math..

[16]  Ladnor Geissinger.  Hopf algebras of symmetric functions and class functions , 1977 .

[17]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .