When is the Bloch-Okounkov q-bracket modular?

Abstract: We obtain a condition describing when the quasimodular forms given by the Bloch-Okounkov theorem as $q$-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator {\Delta}. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.