Families of quasimodular forms arise naturally in many situations such as
curve counting on Abelian surfaces and counting ramified covers of orbifolds.
In many cases the family of quasimodular forms naturally arises as the
coefficients of a Taylor expansion of a Jacobi form. In this note we give
examples of such expansions that arise in the study of partition statistics.
The crank partition statistic has gathered much interest recently. For
instance, Atkin and Garvan showed that the generating functions for the moments
of the crank statistic are quasimodular forms. The two variable generating
function for the crank partition statistic is a Jacobi form. Exploiting the
structure inherent in the Jacobi theta function we construct explicit
expressions for the functions of Atkin and Garvan. Furthermore, this
perspective opens the door for further investigation including a study of the
moments in arithmetic progressions. We conduct a thorough study of the crank
statistic restricted to a residue class modulo 2.Comment: 11 pages. many minor corrections made from previous versio