Computational Methods in Permutation Patterns

OF THE DISSERTATION Computational Methods in Permutation Patterns by Brian Koichi Nakamura Dissertation Director: Doron Zeilberger Given two permutations σ ∈ Sk and π = π1 . . . πn ∈ Sn, the permutation π is said to contain the pattern σ if there exists 1 ≤ i1 < . . . < ik ≤ n such that πi1 . . . πik is order-isomorphic to σ. Each such subsequence is called an occurrence of σ in π. Over the past few decades, the study of pattern-avoiding permutations has been a very active area of research. This thesis will consider two types of problems in this area. The first is a variation known as consecutive patterns, where the pattern σ must occur in consecutive terms of the permutation π to count as an occurrence. The second is a generalization of the classical pattern avoiding problem, where we wish to study permutations with exactly r occurrences of a pattern (for some fixed r ≥ 0).