Combinatorics of Orbit Configuration Spaces

From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this “orbit configuration space” is the complement of an arrangement of subvarieties inside the Cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement), which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations.

Asymptotic normality of coefficients of some polynomials related to Dowling lattices

Recently, we introduced two sequences of polynomials (Bn(x, y, z)) and (Fn(x, y, z)), which unify many familiar polynomials related to Dowling lattices, such as the Bell polynomials, the Dowling polynomials, the ordered Bell polynomials, the r-Bell polynomials and the r-Dowling polynomials. In this paper, we show the asymptotic normality of coefficients of Bn(x, y, z) and Fn(x, y, z). As applications, we obtain the asymptotic normality of coefficients of some polynomials related to Dowling lattices in a unified approach.

Plethysm for Wreath Products and Homology of Sub-Posets of Dowling Lattices

We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.