Traces of partition Eisenstein series

We study "partition Eisenstein series", extensions of the Eisenstein series G₂ₖ(τ), defined by λ=(1m₁, 2m₂,, kmₖ) k \ \ \ \ \ \ \ \ \ \ Gλ(τ):= G₂(τ)m₁ G₄(τ)m₂⋯ G₂ₖ(τ)mₖ. For functions φ: P→ C on partitions, the weight $2k$ "partition Eisenstein trace" is the quasimodular form Trₖ(φ;τ):=∑λ k φ(λ)Gλ(τ). These traces give explicit formulas for some well-known generating functions, such as the kth elementary symmetric functions of the inverse points of 2-dimensional complex lattices Z⊕ Zτ, 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.