Partitions associated to class groups of imaginary quadratic number fields (Q2697662)

A partition \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)\) of an integer \(n\) is called attainable if \(\lambda_1 \geq \lambda_2 + 3\lambda_3 + 5\lambda_4 + \cdots + (2k - 3)\lambda_k.\) Recently, \textit{S. Holmin} et al. [Exp. Math. 28, No. 2, 233--254 (2019; Zbl 1470.11285)] showed that an attainable partition of an integer corresponds to a family of abelian \(p\)-groups that conjecturally are realized as the class groups of imaginary quadratic number fields for infinitely many odd primes \(p\). In this paper, the authors investigate some properties of attainable partitions and establish connection to partitions into triangular numbers.