Restricted overpartitions and concave compositions: their modularity and asymptotics

TL;DR

The paper analyzes restricted overpartitions and concave compositions, revealing that their generating functions sit at the intersection of modular forms, mock theta functions, false theta functions, and mock Maass theta functions. It develops a comprehensive suite of $q$-series tools and Tauberian methods to extract precise asymptotics for associated partition counts, establishing explicit main-term formulas and monotonicity results. By deriving new representations and two-variable generalizations, the work connects these combinatorial objects to Ramanujan-type identities and mixed mock modular phenomena, including a Ramanujan AB5 identity and two-variable mock Maass theta structures. The results advance the modular-analytic understanding of partition statistics and provide a framework for further exploration of rank statistics and restricted partitions in the context of modern automorphic theory.

Abstract

In this paper we study restricted overpartitions and concave compositions. Using q-series transformations, we show that their generating functions are related to modular forms, mock theta functions, false theta functions, and mock Maass theta functions. Moreover, we obtain their asymptotic main terms. We also study related rank statistics.

Theorems & Definitions (50)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Corollary 1.10