A crank-based approach to the theory of 3-core partitions

This note is concerned with the set of integral solutions of the equation $x^2+3y^2=12n+4$, where $n$ is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the 3-core partitions of n. As a consequence we describe an explicit bijection between $3$-core partitions and ideals of the ring of Eisenstein integers, explaining a result of G. Han and K. Ono obtained using modular forms.