ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

How 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.

The labelled-to-labelled cells of the twelvefold way at n=3 and k=2

Example

For $n = 3$ labelled balls and $k = 2$ labelled boxes:

arbitrary maps: $2^{3} = 8$ ;
injective maps: $0$ , because $3 > 2$ ;
surjective maps: $2! S (3, 2) = 2 \cdot 3 = 6$ .

So the labelled-to-labelled row is $(8, 0, 6)$ .

Depends on

The twelvefold way

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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.