Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

Kathleen Petersen; James Sellers

paper

Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

arXiv:2111.12031

Abstract

We investigate properties of attainable partitions of integers, where a partition $(n_{1}, n_{2}, \dots, n_{r})$ of $n$ is attainable if $\sum (3 - 2 i) n_{i} \geq 0$ . Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian $p$ -groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes $p$ . We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.