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 , is a mixed mock modular form in a more general sense than is typically used.
We relate 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 -series manipulations and holomorphic projection.
As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of . For comparison, we give an asymptotic expansion for the number of concave compositions of with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.
"Modularity of the concave composition generating function." Algebra Number Theory 7 (9) 2103 - 2139, 2013. https://doi.org/10.2140/ant.2013.7.2103