DefinitionDefinition: Literature-sourcedProof: Not applicableSession-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.

Set partitions and blocks

Definition

Let $A$ be a set. A set partition of $A$ is a family $P$ of nonempty subsets of $A$ such that:

distinct members of $P$ are disjoint; and
$⋃ P = A$ .

The members of $P$ are its blocks.

Equivalently, a set partition is a decomposition of $A$ into nonempty pairwise disjoint pieces. The empty set has one partition, namely the empty family.

Used by

Two set partitions can have the same block-size multiset and still be different Counterexample
The Stirling numbers of the second kind and the Bell numbers Definition
The set partitions of a finite set form a finite set Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Herbert S. Wilf, generatingfunctionology, 2nd ed., §3.6 (standard reference, not scraped)
Andrew Lin, 18.212 Algebraic Combinatorics, Lecture 11 (standard reference, not scraped)