#### Vol. 7, No. 9, 2013

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Modularity of the concave composition generating function

### George E. Andrews, Robert C. Rhoades and Sander P. Zwegers

Vol. 7 (2013), No. 9, 2103–2139
##### 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 $v\left(q\right)$, is a mixed mock modular form in a more general sense than is typically used.

We relate $v\left(q\right)$ 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.

##### Keywords
concave composition, partition, unimodal sequences, mock theta function, mixed mock modular form
##### Mathematical Subject Classification 2010
Primary: 05A17
Secondary: 11P82, 11F03
##### Milestones
Revised: 10 September 2012
Accepted: 22 October 2012
Published: 18 December 2013
##### Authors
 George E. Andrews Department of Mathematics The Pennsylvania State University 109 McAllister Building University Park PA 16802-7000 United States Robert C. Rhoades Department of Mathematics Stanford University Bldg 380 Stanford CA 94305 United States Sander P. Zwegers Mathematical Institute University of Cologne Weyertal 86-90 D-50931 Cologne Germany