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.

$S (4, 2)$ computed by recurrence, inclusion-exclusion, and direct listing

Example

Three standard computations agree:

recurrence: $S (4, 2) = 2 S (3, 2) + S (3, 1) = 2 \cdot 3 + 1 = 7;$
inclusion-exclusion: $S (4, 2) = \frac{1}{2!} (2^{4} - 2 \cdot 1^{4}) = \frac{14}{2} = 7;$
direct listing: The set partitions of $[4]$ grouped by their number of blocks shows exactly seven two-block partitions of $[4]$ .

Thus all three routes give the same value $S (4, 2) = 7$ .

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

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

S(4,2)S(4,2) computed by recurrence, inclusion-exclusion, and direct listing