k-Run Overpartitions and Mock Theta Functions

Kathrin Bringmann; Alexander Holroyd; Karl Mahlburg; Masha Vlasenko

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arXiv · 1203.6344

k-Run Overpartitions and Mock Theta Functions

Kathrin Bringmann·Alexander Holroyd·Karl Mahlburg·Masha Vlasenko

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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.

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BibTeX RIS

Kathrin Bringmann, Alexander Holroyd, Karl Mahlburg, Masha Vlasenko. 2012-03-28. k-Run Overpartitions and Mock Theta Functions. https://arxiv.org/abs/1203.6344

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