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 involutions of $[4]$ and their cycle decomposition

Example

The involutions of $[4]$ are exactly the permutations whose cycle lengths are all $1$ or $2$ :

one with four fixed points: $()$ ;
six with one transposition: $(12)$ , $(13)$ , $(14)$ , $(23)$ , $(24)$ , $(34)$ ;
three with two disjoint transpositions: $(12) (34)$ , $(13) (24)$ , $(14) (23)$ .

Hence $I_{4} = 1 + 6 + 3 = 10$ , agreeing with the EGF $\exp (x + x^{2} / 2)$ from Standard labelled specializations give involutions, ordered Bell numbers, and partitions without singletons.

Depends on

Standard labelled specializations give involutions, ordered Bell numbers, and partitions without singletons

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)

The involutions of [4][4] and their cycle decomposition