Automating the enumeration of pattern-avoiding permutations

Abstract: When a combinatorial specification is known for a combinatorial class there are well-established tools, such as symbolic and analytic combinatorics, that derive an enumeration, asymptotics, and the ability to randomly generate instances of the objects. However, the initial step from definition to structural description is often ad-hoc. This is the gap combinatorial exploration attempts to fill.

In this seminar, we will focus on the TileScope algorithm, an application of combinatorial exploration to permutation classes. It is capable of finding a combinatorial specification for almost every permutation class that avoids some subset of length 4 patterns, and many more. The enumerations of these classes were either unknown or were the main results of research papers, some as recent as 2018. We will discuss the strategies required for these specifications.

Mathematics

Audience: researchers in the topic

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Algebraic and enumerative combinatorics seminar