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A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

  • DANIEL M. KANE (a1) and ROBERT C. RHOADES (a2)

Abstract

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of $n$ without $k$ consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.

Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without $k$ consecutive parts. Andrews showed that when $k=2$ , the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For $k>2$ , the asymptotic properties of the generating functions have proved more difficult to obtain. Using $q$ -series identities and the $k=2$ case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case $k=3$ was given by Zagier.

This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[22] Zagier, D., ‘Ramanujan’s mock theta functions and their applications [d’aprés Zwegers and Bringmann-Ono]’, inSém. Bourbaki (2007/2008), Astérisque, No. 326, Exp. No. 986, vii–viii (2010), 143164.
[23] Zagier, D., private communication.
[24] Zwegers, S., ‘Mock theta functions’, PhD Thesis (Advisor: D. Zagier), Universiteit Utrecht, (2002).
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A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

  • DANIEL M. KANE (a1) and ROBERT C. RHOADES (a2)

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