### Abstract

A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted v(q), is a mixed mock modular form in a more general sense than is typically used. We relate v(q) to generating functions studied in connection with "Moonshine of the Mathieu group" and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as q-series manipulations and holomorphic projection. As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of n. For comparison, we give an asymptotic expansion for the number of concave compositions of n with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.

Original language | English (US) |
---|---|

Pages (from-to) | 2103-2139 |

Number of pages | 37 |

Journal | Algebra and Number Theory |

Volume | 7 |

Issue number | 9 |

DOIs | |

State | Published - Dec 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Algebra and Number Theory*,

*7*(9), 2103-2139. https://doi.org/10.2140/ant.2013.7.2103

}

*Algebra and Number Theory*, vol. 7, no. 9, pp. 2103-2139. https://doi.org/10.2140/ant.2013.7.2103

**Modularity of the concave composition generating function.** / Andrews, George E.; Rhoades, Robert C.; Zwegers, Sander P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Modularity of the concave composition generating function

AU - Andrews, George E.

AU - Rhoades, Robert C.

AU - Zwegers, Sander P.

PY - 2013/12/1

Y1 - 2013/12/1

N2 - A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted v(q), is a mixed mock modular form in a more general sense than is typically used. We relate v(q) to generating functions studied in connection with "Moonshine of the Mathieu group" and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as q-series manipulations and holomorphic projection. As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of n. For comparison, we give an asymptotic expansion for the number of concave compositions of n with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.

AB - A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted v(q), is a mixed mock modular form in a more general sense than is typically used. We relate v(q) to generating functions studied in connection with "Moonshine of the Mathieu group" and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as q-series manipulations and holomorphic projection. As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of n. For comparison, we give an asymptotic expansion for the number of concave compositions of n with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.

UR - http://www.scopus.com/inward/record.url?scp=84892712007&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892712007&partnerID=8YFLogxK

U2 - 10.2140/ant.2013.7.2103

DO - 10.2140/ant.2013.7.2103

M3 - Article

VL - 7

SP - 2103

EP - 2139

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 9

ER -