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The Combinatorics of the Permutation Enumeration of Wreath Products between Cyclic and Symmetric Groups


Jennifer D. Wagner

Brenti introduced a homomorphism $\xi:\Lambda \rightarrow Q(x)$ defined on the the elementary symmetric functions by

$\xi(e_k(X)) = \frac{(1-x)^k}{k!}$

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 well-known 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.

  • We develop the representation theory of $C_kxS_n$ in an appropriate way, including the definition of a characteristic map from the class functions on $C_kxS_n$ to a space of symmetric functions, and an extension of lambda-ring notation to take into account the complex signs.

  • We determine combinatorial interpretations of the transition matrices between bases of the appropriate space.

  • We define appropriate statistics on the elements or $C_kxS_n$. Since there are a number of ways to define such statistics, we are forced to choose among several possible definitions.

  • We use combinatorial methods to define an analog of $\xi$, which when applied to certain basis elements, gives the desired generating functions on elements of $C_kxS_n$.

  • We give combinatorial proofs of the desired identities. The proofs include interpretation of sums in terms of combinatorial objects, and the performance of involutions on the objects.

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