UC San Diego

Jones and Remmel developed the Reciprocity Method to study the generating functions for the number of descents and left-to-right minima in permutations which have no consecutive tau-matches where tau is a permutation that starts with 1 and has exactly one descent. In this thesis, we extend the reciprocity method to the case of family of permutations Gamma such that, for each permutations tau in Gamma, tau starts with 1 but there is no restrictions on the number of descents in tau. In addition, we also obtain the q-analog for the reciprocity method which leads to many natural refinements of the c-Wilf equivalence relation.