Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 9 (2013), 2103-2139.
Modularity of the concave composition generating function
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 , is a mixed mock modular form in a more general sense than is typically used.
We relate 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 -series manipulations and holomorphic projection.
As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of . For comparison, we give an asymptotic expansion for the number of concave compositions of with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.
Algebra Number Theory, Volume 7, Number 9 (2013), 2103-2139.
Received: 30 July 2012
Revised: 10 September 2012
Accepted: 22 October 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Andrews, George; Rhoades, Robert; Zwegers, Sander. Modularity of the concave composition generating function. Algebra Number Theory 7 (2013), no. 9, 2103--2139. doi:10.2140/ant.2013.7.2103. https://projecteuclid.org/euclid.ant/1513730089