On generalized Schur’s partitions

Abstract

In 1926, I. Schur proved that the number of partitions of [Formula: see text] into parts [Formula: see text] equals the number of partitions of [Formula: see text] of the form [Formula: see text] such that [Formula: see text] with strictly inequality if [Formula: see text]. We prove that Schur’s partition function is related to a weight 3/2 Hecke eigenform modulo 2. As a consequence, we obtain some congruences for Schur’s partitions functions. Schur’s partitions theorem can be easily generalized to general moduli. We show that the generalized Schur’s partition functions satisfy beautiful transformation properties. We also find an unexpected relation between the generalized Schur’s partitions and singular overpartitions, which was introduced by G. E. Andrews recently.