Partition number identities which are true for all set of parts

Abstract

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a restriction $\alpha$ of the number of equal parts. Although there are many way of the expression, we prove that there exists a expression form such that this expression form is true for all possible set $B$. This identities comes from the partition numbers of natural numbers into $\{1,\alpha,\alpha^2,\alpha^3,\cdots\}$. Furthermore, we prove that there exist inverse forms of the expression forms. And we prove other similar identities. The proofs in this paper are constructive.