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.

Two set partitions can have the same block-size multiset and still be different

Statement refuted

Refuted claim: a set partition is determined by the multiset of its block sizes.

Facts & Assumptions

Given: The partitions $P = {{1, 2}, {3, 4}}$ and $Q = {{1, 3}, {2, 4}}$ of $[4]$ .

Proof

technique · direct

1.1

Both $P$ and $Q$ are set partitions of $[4]$ , and both have block-size multiset ${2, 2}$ .

given

2.1

But ${1, 2}$ is a block of $P$ and not of $Q$ , so $P \neq Q$ . Thus the same block sizes do not determine the partition.

step 1.1

3.1

Therefore the claim is false.

step 2.1∎

Depends on

Set partitions and blocks

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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., §3.6 (standard reference, not scraped)