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 $v\left(q\right)$, is a mixed mock modular form in a more general sense than is typically used.

We relate $v\left(q\right)$ 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.