#### Date of Award

12-2005

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

Carl G. Wagner

#### Committee Members

John Nolt, Jan Rosinski, Pavlos Tzermias

#### Abstract

A *q*-generalization *G n*(

*q*) of a combinatorial sequence

*G*which reduces to that sequence when

*n**q*= 1 is obtained by

*q*-counting a statistic defined on a sequence of finite discrete structures enumerated by

*G*. In what follows, we evaluate

*n**G*(

*n**−*1) for statistics on several classes of discrete structures, giving both algebraic and combinatorial proofs. For the latter, we define appropriate sign-reversing involutions on the associated structures. We shall call the actual algebraic result of such an evaluation at

*q*=

*−*1 a

*parity*theorem

*(for the statistic on the associated class of discrete structures).*Among the structures we study are permutations, binary sequences, Laguerre configurations, derangements, Catalan words, and finite set partitions.

As a consequence of our results, we obtain bijective proofs of congruences involving Stirling, Catalan, and Bell numbers. In addition, we modify the ideas used to construct the aforementioned sign-reversing involutions to furnish bijective proofs of combinatorial identities involving sums with alternating signs.

#### Recommended Citation

Shattuck, Mark A., "Parity Theorems for Combinatorial Statistics. " PhD diss., University of Tennessee, 2005.

https://trace.tennessee.edu/utk_graddiss/2327