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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2018
- DOI:
- arXiv:
- arXiv:1803.08095
- Bibcode:
- 2018arXiv180308095K
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- The theorems of this paper stated and proved in 2012 winter