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

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.

concave composition, partition, unimodal sequences, mock theta function, mixed mock modular form
Mathematical Subject Classification 2010
Primary: 05A17
Secondary: 11P82, 11F03
Received: 30 July 2012
Revised: 10 September 2012
Accepted: 22 October 2012
Published: 18 December 2013
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