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.

Ordered Bell numbers already differ from Bell numbers at $n = 3$

Example

For $[3]$ , the Bell number is

$B_{3} = S (3, 1) + S (3, 2) + S (3, 3) = 1 + 3 + 1 = 5.$

The ordered Bell number is larger because each partition with $k$ blocks can be ordered in $k!$ ways:

$1 \cdot 1! + 3 \cdot 2! + 1 \cdot 3! = 1 + 6 + 6 = 13.$

So the ordered Bell sequence already differs from the Bell sequence at $n = 3$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.8 (standard reference, not scraped)

Ordered Bell numbers already differ from Bell numbers at n=3n=3