UC San Diego

The central theme of this dissertation is to develop methods for extending results on generating functions for permutation statistics in several different ways. For example, in Chapter 2, we shall find the generating function analogous to Carlitz's, except that we extend the results to sum over permutations that contain a given finite descent set. In essence, we are forcing the permutations in our sum to have descents at given positions. This result requires proving new identities for ribbon Schur functions. In Chapter 3, we extend the work of Mendes on k-alternating permutations to start and end with any number of elements mod k. In particular, we define a new class of symmetric functions and associated identities to achieve this result. In Chapter 4, we combine the results of Chapters 2 and 3 to permutations with repeating descent positions, except we force rises at a finite number of places. In Chapter 5, we again extend the results of Remmel and Mendes to more general patterns, by which we mean those permutations which have repeating patterns of descents and rises. In order to do this, we must modify the homomorphisms used previously and also modify the labeling system in this machinery. We also present a some combinatorial results on bases for the space of symmetric functions. In Chapter 6, we find a combinatorial interpretation for the coefficients of the dual basis to the ribbon Schur functions indexed by partitions expanded in terms of the monomial symmetric functions