#### Year of Publication

2015

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Richard Ehrenborg

#### Abstract

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their **cd**-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two **cd**-monomials. Then the **cd**-index is defined for infinite posets, with the calculation of the **cd**-index of the universal Coxeter group under the Bruhat order as an example. Finally, an extension of the Pfaffian of a skew-symmetric function, called the hyperpfaffian, is given in terms of a signed sum over partitions of n elements into blocks of equal size. Using a sign-reversing involution on a set of weighted, oriented partitions, we prove an extension of Torelli's Pfaffian identity that results from applying the hyperpfaffian to a skew-symmetric polynomial.

#### Recommended Citation

Fox, Norman B., "Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians" (2015). *Theses and Dissertations--Mathematics*. 25.

https://uknowledge.uky.edu/math_etds/25