Stable multivariate WW-Eulerian polynomials

We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type $B$ is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types $A$ and $C$. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type $D$, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types $B$ and $D$, we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.

ENUMERATING PROJECTIVE REFLECTION GROUPS

Projective re ection groups have been recently dened by the second author. They include a special class of groups denoted G(r; p; s; n) which contains all classical Weyl groups and more generally all the complex re ection groups of type G(r; p; n). In this paper we dene some statistics analogous to descent number and major index over the projective re ection groups G(r; p; s; n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r; p; s; n), as distribution of one-dimensional characters and computation of Hilbert series of invariant algebras, are also treated

Specializations of colored quasisymmetric functions and Euler-Mahonian identities

We propose a unified approach to prove general formulas for the joint distribution of an Eulerian and a Mahonian statistic over a set of colored permutations by specializing Poirier's colored quasisymmetric functions. We apply this method to derive formulas for Euler-Mahonian distributions on colored permutations, derangements and involutions. A number of known formulas are recovered as special cases of our results, including formulas of Biagioli-Zeng, Assaf, Haglund-Loehr-Remmel, Chow-Mansour, Biagioli-Caselli, Bagno-Biagioli, Faliharimalala-Zeng. Several new results are also obtained. For instance, a two-parameter flag major index on signed permutations is introduced and formulas for its distribution and its joint distribution with some Eulerian partners are proven.Comment: 36 pages, no figure