Modularity of the concave composition generating function

Andrews, George; Rhoades, Robert; Zwegers, Sander

doi:10.2140/ant.2013.7.2103

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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

DOI: 10.2140/ant.2013.7.2103

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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 05A17

Secondary: 11P82, 11F03

Milestones

Received: 30 July 2012

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

Vol. 7, No. 9, 2013