A combinatorial interpretation of the probabilities of p-groups in the Cohen-Lenstra measure

In this paper, I will introduce a link between the volume of a finite abelian p-group in the Cohen-Lenstra measure and partitions of a certain type. These partitions will be classified by the output of an algorithm. As a corollary, I will give a formula (7.2) for the probability of a p-group to have a specific exponent.