[Dissertation] A polynomial invariant of partial orders

Biographical Sketch
Dedication
Acknowledgements
Table of Contents
List of Figures
Introduction
Preliminaries
	Species and Hopf Monoids
		Set Species
		Monoids and Comonoids in the Category of Vector Species
		Bimonoids
		Set Species, The Linearization Functor, and the Antipode
		Characters and Polynomial Invariants
		The Hopf Monoid of Partial Orders
	Hyperplane Arrangements and A-species
		Hyperplanes and Hyperplane Arrangements
		Faces and Chambers
		Sign Sequences and the Tits Product
		Flats
		The Braid Arrangement
		A-species
		Monoids, Comonoids, and Bimonoids in A-species
		The Antipode for A-Hopf monoids
The Antipode Formula for the Hopf Monoid of Partial Orders
	p-Compatible Compositions of Partial Orders
	The Antipode of Partial Orders.
	The Antipode Formula Indexed by Compositions
The Antipode Formula Generalization to Simplicial Hyperplane Arrangements
	Cones, Top-Cones, Lunes and Wedges
	The A-Hopf monoid of Top-Cones.
	Generalizing from Posets
	The Antipode of the A-Hopf monoid of Top-Cones.
Characters From The Width of a Partial Order
An Invariant from the Character of Linear Extensions of a Partial Order
	The Polynomial Invariant
	The Approach
	Computing e-1(p)k; X(p,k) for k Reverse Linear
		The Failure of Conjecture 52
	Characterizing X(p,k) and X(p,k)c for all linear orders
	Proof of Conjecture 42
	Proof of Theorem 76
	The Polynomial of Dorpalen-Barry, Kim, and Reiner
A Family of Simplicial Complexes
	From X(p,k)c to the Simplicial Complex of a NonNesting Interval Set
	The Final Nerve Construction
	Proof of Conjecture 51
	A Note on Final Nerves of Finite Simplicial Complexes
Bibliography