Starting with a counting problem for elements of the symmetric group, we introduce the so-called shifted symmetric functions. These functions, which also occur naturally in enumerative geometry, have the remarkable property that the corresponding generating series are quasimodular forms. We discuss another family of functions on partitions with the same property. In particular, using certain Hamiltonian operators associated to cohomological field theories, we explain how this seemingly different family of functions turns out to be closely related to the shifted symmetric functions.