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.

There are 11 S_4-orbits of edge-sets on the pairs of [4]

Example

For the action of $S_{4}$ on the $6$ unordered pairs of $[4]$ , the induced cycle-index polynomial is

$\frac{1}{24} (s_{1}^{6} + 9 s_{1}^{2} s_{2}^{2} + 8 s_{3}^{2} + 6 s_{2} s_{4}) .$

Substituting $s_{i} = 1 + x^{i}$ gives

$1 + x + 2 x^{2} + 3 x^{3} + 2 x^{4} + x^{5} + x^{6} .$

In particular, setting $x = 1$ yields

$1 + 1 + 2 + 3 + 2 + 1 + 1 = 11,$

so there are $11$ edge-set orbits, equivalently $11$ simple graphs on $4$ vertices up to isomorphism.

Depends on

Pólya enumeration counts edge-set orbits on the 2-subsets of [n]

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.