On derangement polynomials of type B. II

We define a type B derangement polynomial by q-counting derangements by the number of excedances in the hyperoctahedral group B"n. Properties of the derangement polynomial, including the recurrence relation, generating function, real-rootedness, reciprocity and a unimodal decomposition arising from a symmetric function identity, are studied. The results presented generalize and unify earlier ones due to Brenti, Stanley, Stembridge and Zhang.

[1]  Yuval Roichman,et al.  The Flag Major Index and Group Actions on Polynomial Rings , 2001, Eur. J. Comb..

[2]  Yuval Roichman,et al.  Descent Numbers and Major Indices for the Hyperoctahedral Group , 2000, Adv. Appl. Math..

[3]  R. Stanley.  Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a , 1989 .

[4]  F. Brenti,et al.  Unimodal, log-concave and Pólya frequency sequences in combinatorics , 1989 .

[5]  Michael J. Mossinghoff,et al.  Combinatorics and graph theory , 2000 .

[6]  Zhang Xiangde.  On a Kind of Sequence of Polynomials , 1995, COCOON.

[7]  Francesco Brenti,et al.  q-Eulerian Polynomials Arising from Coxeter Groups , 1994, Eur. J. Comb..

[8]  Guo-Niu Han,et al.  Signed Words and Permutations II; The Euler-Mahonian Polynomials , 2005, Electron. J. Comb..

[9]  I. Gessel,et al.  Permutation statistics and partitions , 1979 .

[10]  Ira M. Gessel,et al.  On the descent numbers and major indices for the hyperoctahedral group , 2007, Adv. Appl. Math..

[11]  Francesco Brenti,et al.  Unimodal polynomials arising from symmetric functions , 1990 .

[12]  Michelle L. Wachs,et al.  On $q$-derangement numbers , 1989 .

[13]  John R. Stembridge,et al.  Eulerian numbers, tableaux, and the Betti numbers of a toric variety , 1992, Discret. Math..

[14]  Guo-Niu Han,et al.  Signed words and permutations, III; the MacMahon Verfahren , 2006 .

[15]  Jeffrey B. Remmel,et al.  Statistics on wreath products, perfect matchings, and signed words , 2005, Eur. J. Comb..

[16]  A. Björner,et al.  Combinatorics of Coxeter Groups , 2005 .

[17]  F. Brenti,et al.  Permutation enumeration symmetric functions, and unimodality , 1993 .