Combinatorial approaches to problems about integer partitions and q-series

Burson, Hannah Elizabeth

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https://hdl.handle.net/2142/108575 Copy

Description

Title

Combinatorial approaches to problems about integer partitions and q-series

Author(s)

Burson, Hannah Elizabeth

Issue Date

2020-07-06

Director of Research (if dissertation) or Advisor (if thesis)

Berndt, Bruce C

Doctoral Committee Chair(s)

Reznick, Bruce

Committee Member(s)

• Ahlgren, Scott D
• Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• partitions
• q-series
• mock theta functions
• false theta functions

Abstract

In this thesis, we study three types of partition-theoretic objects. The first objects studied are odd Ferrers diagrams, which we use in new combinatorial proofs of two general q-series identities containing false theta function and mock theta function identities as special cases. Additionally, we introduce a Gaussian coefficient analogue for restricted odd Ferrers diagrams. The second set of objects we study is a class of core partitions. We prove a recursive formula for the generating polynomial for the number of parts of s-core partitions into d-distinct parts with a given maximum hook length. Applications of the main result of this section include formulas for the maximum number of parts of simultaneous s- and t- core partitions for certain values of t. In the final section, we introduce a new class of objects called copartitions, which generalize a special class of partitions where all even parts are smaller than all odd parts. A simple graphical representation immediately follows from our definition. We prove an in finite product generating function for cp(n), the number of copartitions of size n. Special cases of this function include sums of the ordinary partition function p(n), a new combinatorial interpretation of the Rogers-Ramanujan functions, and quotients involving theta functions and eta functions. Furthermore, we explore the parity of cp(n) and consider a weighted count of copartitions.

Graduation Semester

2020-08

Type of Resource

Thesis

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http://hdl.handle.net/2142/108575

Copyright and License Information

Copyright 2020 Hannah E. Burson

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