LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Ranks of the Brauer and partition diagram bases

Statement

For $a \geq 0$ , the Brauer algebra $B_{a} (δ)$ over any commutative unital ring $R$ is free of rank $(2 a - 1)!!$ , with $(- 1)!! = 1$ . The partition algebra $P_{a} (δ)$ is free of rank $Bell (2 a)$ , where $Bell (j)$ denotes the number of set partitions of a $j$ -element set.

Facts & Assumptions

Given: The diagram bases in Typed Brauer and partition diagram categories.

[F1]

Typed Brauer and partition diagram categories: $B_{a} (δ)$ has one free basis vector per perfect matching of $2 a$ ordered vertices; $P_{a} (δ)$ has one per set partition of those vertices.

Proof

1.1givenF1algebra

Let $M_{2 a}$ be the number of perfect matchings of $2 a$ labels. For $a = 0$ the empty matching gives $M_{0} = 1$ . For $a \geq 1$ , the first label can be paired with any of the remaining $2 a - 1$ labels, and deleting that pair leaves an arbitrary matching of $2 a - 2$ labels. Hence $M_{2 a} = (2 a - 1) M_{2 a - 2}$ , and induction gives $M_{2 a} = (2 a - 1) (2 a - 3) \dots 1 = (2 a - 1)!!$ .

2.1F1algebra∎

By [F1], partition diagrams are in bijection with set partitions of the $2 a$ boundary labels. Their number is $Bell (2 a)$ by the definition in the Statement, and the module is free on precisely that set.

Depends on

Typed Brauer and partition diagram categories

Used by

Dependency tree · one level

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Sources

Lehrer and Zhang, The Brauer Category and Invariant Theory, §2.1 (standard reference, not scraped)
Halverson and Ram, Partition Algebras, §2 (standard reference, not scraped)