Lambert series and conjugacy classes in GL

A relationship between the general linear group GL(n,m) and integer partitions was investigated by Macdonald in order to calculate the number of conjugacy classes in GL(n,m). In this paper, the author introduced two different factorizations for a special case of Lambert series in order to prove that the number of conjugacy classes in the general linear group GL(n,m) and the number of partitions of n into k different magnitudes are related by a finite discrete convolution. New identities involving overpartitions, partitions into k different magnitudes and other combinatorial objects are discovered and proved in this context.