Date of Completion

8-10-2020

Embargo Period

8-10-2020

Keywords

Homomesy, Foata, Permutations

Major Advisor

Tom Roby

Associate Advisor

David Reed Solomon

Associate Advisor

Ralf Schiffler

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In this thesis, we consider two different families of maps on the symmetric group Sn, each created by intertwining a bijection of Foata with dihedral involutions on permutation matrices. Iterating each map creates a cyclic action on Sn, partitioning it into orbits. This allows us to look at statistics that have the same average value over each orbit, called homomesic. The homomesy phenomenon was first proposed by Propp and Roby in 2011, and many instances have been found across a wide range of combinatorial objects and maps.

The first family of maps involves the so-called “fundamental bijection” of R ́enyi and Foata, which “drops parentheses” from a permutation in canonical disjoint cycle decomposition. The second, due to Foata and Schu ̈tzenberger, was originally used to provide a bijective proof showing the equidistribution across Sn of the inversion number and the major index. Computations in SageMath led to a number of con- jectural homomesies on well-known permutation statistics. We prove many of them here, and state the remainder as open problems.

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