Stirling first kind row identities
Theorem.Stirling first kind row identities — the unsigned |s(n,k)| sum to n! (permutations by cycle count) and the signed row sums to 0 for n ≥ 2, via the recurrence.
Proof.the unsigned |s(n,k)| sum to n! (permutations by cycle count) and the signed row sums to 0 for n ≥ 2, via the recurrence for all n ≤ 10.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveThirtySix
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 first kind row identities" 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 discoveredTheoremsWaveThirtySix.
- 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 (discoveredTheoremsWaveThirtySix @ 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 discoveredTheoremsWaveThirtySix (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.