UC San Diego

Well-known statistics on the symmetric group include descents, inversions, major index, and the alternating numbers. This dissertation extends these statistics to two classes of wreath product groups. In both classes we study several different definitions of descent and prove generating functions that rely on these definitions. We also see that equivalent definitions of alternation in the symmetric group lead to different generating functions in the wreath product. We prove our results by applying ring homomorphisms to well-known symmetric function identities and interpreting the resulting expressions combinatorially.