Analytic and Combinatorial study of Schur's Second Partition Theorem

The famous Rogers-Ramanujan identities state that the integer partitions of n into distinct non-consecutive parts are equinumerous with the partitions of n into parts congruent to 1 or 4 modulo 5. The analytic statement of this result is a "sum-product" identity that equates a hypergeometric q-series to an infinite product, and has deep connections to vertex operator algebras, representation theory, mathematical physics, and the theory of modular forms. In this talk I will discuss recent results arising from the study of Schur's Second partition theorem (and its generalizations), which similarly relates partitions into distinct parts with certain restrictions to an infinite product. I will briefly describe the analytic perspective and the intrinsic connections to automorphic forms. The first result answers a conjecture of Andrews by providing fundamental identities between Schur's partitions and Hickerson's universal mock theta function. As an application, this allows the use of Wright's Circle Method in order to calculate asymptotic enumeration formulas for Schur's partitions, which also answers the second part of Andrews' speculation. I will also discuss combinatorial aspects of Schur's partition function, and present a new double summation hypergeometric $q$-series representation. In fact, there are several different proof techniques available, including bijective mappings and modular diagrams, the theory of $q$-difference equations and recurrences, and the theories of summation and transformation for $q$-series. These techniques can also be used for the partition identities of Gollnitz and Gordon. If time permits, I will also discuss a probabilistic application of Schur's partitions, which has the striking result that the universal mock theta function can be expressed as a conditional probability in a certain natural probability space with an infinite sequence of independent events.