Combinatorial approaches to problems about integer partitions and q-series
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.
partitions, q-series, mock theta functions, false theta functions
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign