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2013 Modularity of the concave composition generating function
George Andrews, Robert Rhoades, Sander Zwegers
Algebra Number Theory 7(9): 2103-2139 (2013). DOI: 10.2140/ant.2013.7.2103


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.


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George Andrews. Robert Rhoades. Sander Zwegers. "Modularity of the concave composition generating function." Algebra Number Theory 7 (9) 2103 - 2139, 2013.


Received: 30 July 2012; Revised: 10 September 2012; Accepted: 22 October 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1282.05016
MathSciNet: MR3152010
Digital Object Identifier: 10.2140/ant.2013.7.2103

Primary: 05A17
Secondary: 11F03 , 11P82

Keywords: concave composition , mixed mock modular form , mock theta function , Partition , unimodal sequences

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 9 • 2013
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