#### Year of Publication

2017

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department/School/Program

Mathematics

#### First Advisor

Dr. Richard Ehrenborg

#### Abstract

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-elements

#### Digital Object Identifier (DOI)

https://doi.org/10.13023/ETD.2017.191

#### Recommended Citation

Hedmark, Dustin g., "The Partition Lattice in Many Guises" (2017). *Theses and Dissertations--Mathematics*. 48.

https://uknowledge.uky.edu/math_etds/48

#### Included in

Algebra Commons, Discrete Mathematics and Combinatorics Commons, Geometry and Topology Commons