Traces of partition Eisenstein series

Abstract

We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(\tau),$ defined by $$\lambda=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_{\lambda}(\tau):= G_2(\tau)^{m_1} G_4(\tau)^{m_2}\cdots G_{2k}(\tau)^{m_k}. $$ For functions $\phi: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form $$ {\mathrm{Tr}}_k(\phi;\tau):=\sum_{\lambda \vdash k} \phi(\lambda)G_{\lambda}(\tau). $$ These traces give explicit formulas for some well-known generating functions, such as the $k$th elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}\tau,$ as well as the $2k$th power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.