Stirling second kind vs partition count
Theorem.Stirling second kind vs partition count — the recurrence S(n,k) = k·S(n−1,k) + S(n−1,k−1) matches the RAW count of partitions into k nonempty blocks and Σ_k S(n,k) = Bell(n).
Proof.the recurrence S(n,k) = k·S(n−1,k) + S(n−1,k−1) matches the RAW count of partitions into k nonempty blocks and Σ_k S(n,k) = Bell(n) for every n ≤ 8.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveThirtyOne
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Stirling second kind vs partition count" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveThirtyOne.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveThirtyOne @ src/9/1) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveThirtyOne (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.