CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge 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 second-kind inclusion-exclusion formula does not count permutations by cycles

Statement refuted

Refuted claim: the inclusion-exclusion formula for $S (n, k)$ also counts permutations of $[n]$ with exactly $k$ cycles.

Facts & Assumptions

Given: The case $n = 3$ , $k = 1$ .

Proof

technique · direct

1.1

The second-kind formula gives $S (3, 1) = 1$ .

given

2.1

But the permutations of $[3]$ with exactly one cycle are the two $3$ -cycles $(123)$ and $(132)$ , so $c (3, 1) = 2$ . Therefore the second-kind formula does not count permutations by cycles.

step 1.1given

3.1

Hence the claim is false.

step 2.1∎

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)