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

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 π‘Ÿth 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 Ζ’-vector of the descent polytope DPv, proving a maximization result using an analogue of the boustrophedon transform.

Identiferoai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:math_etds-1053
Date01 January 2018
CreatorsHapp, Alexander Thomas
PublisherUKnowledge
Source SetsUniversity of Kentucky
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceTheses and Dissertations--Mathematics

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