Partition number identities which are true for all set of parts

TL;DR

This paper establishes universal identities relating unrestricted and restricted partition numbers for all infinite sets of parts, based on partitions into geometric sequences, with constructive proofs of these identities and their inverses.

Contribution

It introduces a general expression form for partition identities valid for all infinite sets of parts, including identities involving geometric sequences and their inverses.

Findings

Universal partition identities for all infinite sets of parts

Constructive proofs of identities and their inverses

Extensions to similar partition identities

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.