{"id":"e-046150b77085a263","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-849de7332df28657","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"24d7012b69a1765d7ff81b766f5a3e3adc4f306b81a0f38cf8dc1669e646f19b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"definition_equality","certificate_raw_size":6561,"certificate_compressed_size":2062,"conclusion_name":"Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","claims":[],"premises":[],"cited_by":[]}