k-Run Overpartitions and Mock Theta Functions

TL;DR

This paper introduces k-run overpartitions as a new combinatorial object, derives their generating functions, relates 1-run overpartitions to Ramanujan's mock theta functions, and analyzes their asymptotic growth using probabilistic methods.

Contribution

It defines k-run overpartitions, provides their generating functions, connects 1-run overpartitions to mock theta functions, and studies their asymptotic behavior.

Findings

Derived double summation q-hypergeometric series for generating functions.

Connected 1-run overpartitions to Ramanujan's mock theta functions.

Established asymptotic growth estimates for k-run overpartitions.

Abstract

In this paper we introduce k-run overpartitions as natural analogs to partitions without k-sequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of results for k-run overpartitions, beginning with a double summation q-hypergeometric series representation for the generating functions. In the special case of 1-run overpartitions we further relate the generating function to one of Ramanujan's mock theta functions. Finally, we describe the relationship between k-run overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of k-run overpartitions of size n.