UC San Diego

We study the enumeration of combinatorial objects by number of occurrences of patterns and other statistics. This work is broken into three main parts. In the first part, we enumerate permutations, compositions, column- convex polyominoes, and words by patterns relating consecutive entries. We show that there is a hierarchy of enumeration problems on these sets of objects, such that the problems in one set may be reformulated in terms of the higher sets, then solved using powerful techniques developed for those sets. We use this viewpoint to solve an open problem due to Kitaev and to produce many extensions of existing results and interesting new results. In the second part, we use the same viewpoint to generalize a theorem due to Garsia and Gessel on the major index statistic. We give many specializations and slight extensions of this result to apply it to a variety of combinatorial objects and variations of the statistic. In the third part, we present general method for finding bijections between sets of objects that preserve various statistics. We use this method to solve problems posed by Claesson and Linusson and by Jones, and we also present several new results