Algebra & Number Theory

Modularity of the concave composition generating function

George Andrews, Robert Rhoades, and Sander Zwegers

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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.

Article information

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]
Secondary: 11P82: Analytic theory of partitions 11F03: Modular and automorphic functions

concave composition partition unimodal sequences mock theta function mixed mock modular form


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.

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