  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.
