ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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 set partitions of $[4]$ grouped by their number of blocks

Example

The partitions of $[4]$ are:

one block: ${{1, 2, 3, 4}}$ ;
two blocks: ${{1}, {2, 3, 4}}$ , ${{2}, {1, 3, 4}}$ , ${{3}, {1, 2, 4}}$ , ${{4}, {1, 2, 3}}$ , ${{1, 2}, {3, 4}}$ , ${{1, 3}, {2, 4}}$ , ${{1, 4}, {2, 3}}$ ;
three blocks: ${{1}, {2}, {3, 4}}$ , ${{1}, {3}, {2, 4}}$ , ${{1}, {4}, {2, 3}}$ , ${{2}, {3}, {1, 4}}$ , ${{2}, {4}, {1, 3}}$ , ${{3}, {4}, {1, 2}}$ ;
four blocks: ${{1}, {2}, {3}, {4}}$ .

$S (4, 1) = 1, S (4, 2) = 7, S (4, 3) = 6, S (4, 4) = 1,$

and therefore $B_{4} = 1 + 7 + 6 + 1 = 15$ .

Depends on

The Stirling numbers of the second kind and the Bell numbers

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Andrew Lin, 18.212 Algebraic Combinatorics, Lecture 11 (standard reference, not scraped)

The set partitions of [4][4] grouped by their number of blocks