ENUMERATION SCHEMES FOR PATTERN-AVOIDING WORDS AND PERMUTATIONS

Abstract

OF THE DISSERTATION Enumeration Schemes for Pattern-Avoiding Words and Permutations by Lara Kristin Pudwell Dissertation Director: Doron Zeilberger Let p = p1 · · · pn ∈ Sn and q = q1 · · · qm ∈ Sm. We say that p contains q as a pattern if there are indices 1 ≤ i1 < · · · < im ≤ n such that pij < pik ⇐⇒ qi < qk; otherwise, p avoids q. The study of pattern avoidance in permutations is well studied from a variety of perspectives. This thesis is concerned with two generalizations of this pattern avoidance problem. The first generalization is that of pattern avoidance in words (where p and q may have repeated letters). The second is that of barred permutation patterns (where p avoids q unless q is part of an instance of an even larger pattern in p). In both cases, we seek to find universal methods that count words (resp. permutations) avoiding a particular set of patterns, and automate these methods to achieve new enumeration results.