ABSTRACT OF THE DISSERTATION The Combinatorics of the Permutation Enumeration of Wreath Products between Cyclic and Symmetric Groups by Jennifer D. Wagner Brenti introduced a homomorphism $\xi:\Lambda \rightarrow Q(x)$ defined on the the elementary symmetric functions by
where $\Lambda$ is the space of homogeneous polynomials in an infinite number of variables $X = (x_1,x_2, \ldots )$ which are constant under all permutations of these variables. He proved that the homomorphism $\xi$ has the remarkable property that when it is applied to a homogeneous symmetric function $h_k(X)$, the result is the wellknown Eulerian polynomial, which is also the generating function for the number of descents of a permutation. In addition, if $\xi$ is applied to a power symmetric function, the result is a generating function for another permutation statistic. Beck and Remmel used combinatorial interpretations of the transition matrices between bases of $\Lambda$ to give combinatorial proofs of these and other related identities, including $q$analogs. In addition, they used these combinatorial methods to develop an analog of Brenti's permutation enumeration for $B_n$, the hyperoctahedral group consisting of signed permutations. In the dissertation we extend Brenti, Beck and Remmel's results to wreath products $C_kxS_n$ between cyclic and symmetric groups, which can be considered as groups of permutations signed with $k$^{th}$ roots of unity. The key steps in our extension to $C_kxS_n$ include the following.
