Partitions associated to class groups of imaginary quadratic number fields

We investigate properties of attainable partitions integers, where a partition $$(n_1,n_2, \dots , n_r)$$ n is if $$\sum (3-2i)n_i\ge 0$$ . Conjecturally, under an extension the Cohen and Lenstra heuristics by Holmin et. al., these correspond to abelian p-groups that appear as class groups imaginary quadratic number fields for infinitely many odd primes p. demonstrate connection integers into triangular numbers, construct generating function partitions, determine maximal length partitions.