Chak-On Chow

On the Eulerian polynomials of type D

We define and study sub-Eulerian polynomials of type D , which count the elements of the group of even-signed permutations D n with respect to the number of descents in a refined sense. The recurrence relations and exponential generating functions of the sub-Eulerian polynomials are determined, by which the solution to a problem of Brenti, concerning the recurrence relation for the Eulerian polynomials of type D , is also obtained.

Counting signed permutations by their alternating runs

The purpose of this paper is to establish a connection between alternating runs of signed permutations in the hyperoctahedral group and left peaks of permutations in the symmetric group, and then to study some associated enumerative polynomials of signed permutations. Properties of these enumerative polynomials, including combinatorial formulas and recurrence relations are studied. In particular, we deduce a convolution formula connecting the number of alternating runs of permutations in the symmetric group to that in the hyperoctahedral group.

On the real-rootedness of generalized Touchard polynomials

We consider the real-rootedness of generalized Touchard polynomials recently revisited by Mansour and Schork (2013). Towards this end, we first describe the normal form of the generalized Touchard polynomials, by which recurrence relations for the polynomial part are derived. By using the recurrence relations, we prove the real-rootedness of the generalized Touchard polynomials for the parameter m ? 1 , ∞ ) ? k k + 1 : k ? N .