The World of P and Q: Congruences, Identities and Asymptotics
Open Access
- Author:
- Chen, Xiaohang
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 19, 2021
- Committee Members:
- George Andrews, Chair & Dissertation Advisor
Donald Richards, Outside Unit & Field Member
Ae Ja Yee, Major Field Member
Carina Curto, Program Head/Chair
Wen-Ching Li, Major Field Member - Keywords:
- Integer Partitions
q-Series
Congruences
Identities
Asymptotics
Patterns in Inversion Sequences
Linked Partition Ideals
Rogers–Ramanujan Type Identities
Circle Method - Abstract:
- This thesis is devoted to providing some "travel tips" that arise from my personal visit in the world of $p$(artitions) and $q$(-series). In the first part, we will focus on partition congruences, especially from an elementary perspective. We first give a completely elementary proof of an infinite family of congruences modulo powers of 5 for the number of partitions of $n$ into distinct parts. As a by-product, we also consider some eta-quotient representations concerning the Rogers–Ramanujan continued fraction. In the second part, our attention is turned to identities. The first two chapters in this part are devoted to partition identities — one treats weighted partition rank and crank moments and the other investigates partitions with bounded part differences. Then in a series of three chapters, a general theory of span one linked partition ideals will be presented. We start from several conjectures of Kanade and Russell and then link this theory with directed graphs. A comprehensive example on Gleißberg's identity will finally be discussed. The last chapter in this part will be devoted to analytic identities of Rogers–Ramanujan type with manipulations of basic hypergeometric series heavily involved. In the third part, asymptotic aspects of integer partitions will be investigated. We first use square-root partitions into distinct parts to illustrate a refined Meinardus-type approach. In the next three chapters, we will focus on modular infinite products that concern either Dedekind eta function or Jacobi theta function with the assistance of Rademacher's circle method. Finally, we will study nonmodular infinite products that are related to a conjecture of Seo and Yee. In the last part, we will leave for another world of $p$, that is, the world of $p$atterns in inversion sequences. We mainly focus on two recent conjectures, one of which on 0012-avoidance is due to Lin and Ma and the other on the avoidance of triples of binary relations is due to Lin.