Modularity of the concave composition generating function

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 [math] , is a mixed mock modular form in a more general sense than is typically used.\n¶ We relate [math] 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 [math] -series manipulations and holomorphic projection.\n¶ As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of [math] . For comparison, we give an asymptotic expansion for the number of concave compositions of [math] with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.