Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Partitions of a positive integer
Definition
For , a partition of is a finite nondecreasing list of positive integers with , using finite natural sums as in Finite sums and finite products of natural numbers, and in and naturals as in The natural numbers (von Neumann). Equality is equality of these lists. The nondecreasing convention removes permutations from the data.
Depends on
Used by
- Isomorphism classes of abelian groups of order pⁿ are counted by partitions of n Corollary
- Conventions for integer partitions, Ferrers diagrams, and the twelvefold-way table Definition
- Ferrers and Young diagrams, conjugate partitions, self-conjugacy, and the Durfee square Definition
- The functions p(n), pₖ(n), and the standard restricted partition families Definition
- The abelian groups of order 360 in elementary-divisor and invariant-factor form Example
- The five abelian groups of order sixteen Example
Dependency tree · two levels
19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
- Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)