Partitions Associated to Class Groups of Imaginary Quadratic Number Fields

TL;DR

This paper explores the properties of certain integer partitions linked to class groups of imaginary quadratic fields, proposing conjectural connections to number theory heuristics and providing new combinatorial insights.

Contribution

It introduces a new class of partitions related to class groups, constructs a generating function, and analyzes their maximal length, extending understanding of algebraic number theory.

Findings

Connection between attainable partitions and partitions into triangular numbers

Construction of a generating function for attainable partitions

Determination of the maximal length of attainable partitions

Abstract

We investigate properties of attainable partitions of integers, where a partition $(n_1,n_2, \dots, n_r)$ of $n$ is attainable if $\sum (3-2i)n_i\geq 0$. 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.