PhD Dissertation:Yalçın Can Kılıç

A Semi-Constructive Approach To Evidently Positive Partition Generating Functions

Yalçın Can Kılıç
Mathematics PhD Dissertation, 2024

Thesis Jury

Assoc. Prof. Dr Kağan Kurşungöz (Thesis Advisor)

Assoc. Prof. Dr. Mohammad Sadek

Assoc. Prof. Dr. Kamer Kaya

Prof. Dr. Jeremy Lovejoy

Assoc. Prof. Dr. Ayhan Dil

Date & Time: 19th December, 2024 –  14:00

Place: FENS 2072

Zoom Link: https://sabanciuniv.zoom.us/j/92609426391  (meeting id: 926 0942 6391)

Keywords :integer partitions, partition generating functions, evidently positive series, Andrews-Gordon Series, functional equations

Abstract

In this work, we start with the celebrated Rogers–Ramanujan identities, which are fundamental results in the theory of integer partitions. Then, we continue by presenting certain combinatorial generalizations related to these theorems, called Rogers-Ramanujan type identities. These identities involve two types of constraints: modulus constraints and difference constraints. While generating functions on the modulus side are relatively straightforward to construct, manipulate, and interpret, addressing the difference side requires a more nuanced approach.

To this end, we introduce a general framework, termed the moves framework, for interpreting evidently positive series arising from a specific form of two-variable generating functions. This framework is applicable under certain algebraic conditions on the exponents of the generating functions. For cases where these conditions are not satisfied, we propose an alternative method. This involves deriving a system of functional equations satisfied by the series and translating this information into a recursive combinatorial construction, which allows us to provide a combinatorial interpretation of the series.

The thesis concludes with a discussion of potential directions for future research, highlighting open problems and areas for further exploration.