RemarkRemark: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

Why the Catalan count is proved three times, and how the three statements agree

Remarks

The reflection route of $C_{n} + (\binom{2 n}{n + 1}) = (\binom{2 n}{n})$ spends only a path-set bijection: the work is in the first visit to the level $- 1$ , and the final count is the path difference absorbed into $(n + 1) C_{n} = (\binom{2 n}{n})$ .

The cycle-lemma route of $(2 n + 1) C_{n} = (\binom{2 n + 1}{n})$ , a second derivation of the Catalan count spends a free cyclic action and trivial stabilisers. Its conclusion is the cleared count $(2 n + 1) C_{n} = (\binom{2 n + 1}{n})$ , and the last step of that theorem identifies this with the closed form of $(n + 1) C_{n} = (\binom{2 n}{n})$ rather than treating it as a new sequence.

The formal-power-series route of A third derivation of $(n + 1) C_{n} = (\binom{2 n}{n})$ , from the closed form of $C (x)$ spends algebra in $Q ⟦ x ⟧$ : the recurrence becomes the quadratic equation for $C (x)$ , the closed form comes from $2 x C (x) = 1 - (1 - 4 x)^{1 / 2}$ , where $(1 - 4 x)^{1 / 2}$ is the unique square root with constant coefficient $1$ , and coefficient extraction returns the same closed formula again. So the three arguments disagree only in their hypotheses and intermediate objects, not in the count they deliver.

Depends on

Used by

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Dependency tree · two levels

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Sources

A. Postnikov (notes by A. Lin), MIT 18.212 Algebraic Combinatorics, Spring 2019 (standard reference, not scraped)
C. Krattenthaler, "Lattice Path Enumeration", ch. 10 of the Handbook of Enumerative Combinatorics (standard reference, not scraped)
N. Dershowitz and S. Zaks, The Cycle Lemma and Some Applications (standard reference, not scraped)