A Combinatorial Miscellany: Antipodes, Parking Cars, and Descent Set Powers

Author

Alexander Thomas Happ, University of KentuckyFollow

Date Available

4-18-2018

Year of Publication

2018

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department/School/Program

Mathematics

First Advisor

Dr. Richard Ehrenborg

Abstract

In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results depend upon the base 𝑝 expansion of both the dimension and the power of these statistics. Finally, we inspect the ƒ-vector of the descent polytope DP_v, proving a maximization result using an analogue of the boustrophedon transform.

Digital Object Identifier (DOI)

https://doi.org/10.13023/ETD.2018.149

Recommended Citation

Happ, Alexander Thomas, "A Combinatorial Miscellany: Antipodes, Parking Cars, and Descent Set Powers" (2018). Theses and Dissertations--Mathematics. 53.
https://uknowledge.uky.edu/math_etds/53