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 unlabelled-domain to labelled-codomain cells at n=3 and k=2

Example

For $3$ indistinguishable balls and $2$ labelled boxes, the occupancy vectors are:

arbitrary: $(3, 0)$ , $(2, 1)$ , $(1, 2)$ , $(0, 3)$ , so the count is $4$ ;
injective: none, because a $0$ - $1$ vector of length $2$ cannot sum to $3$ ;
surjective: $(2, 1)$ and $(1, 2)$ , so the count is $2$ .

Thus the unlabelled-to-labelled row is $(4, 0, 2)$ .

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.