Proof step e-c4f0d53850ef9f92

basis_map_theorem → concludes about Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0. JSON

Theorem

Index-dependent basis map [Bre89, Thm 2.4.3; MW08, Thm 2] polytool RealRooted basisMap_factorization_of_scalarIndex_recurrence RealRooted rayBasisMap_maps_realRootedIn_iff ?

Operation

Basis map theorem for the node's own rule

{
  "operation": "basis_map_theorem"
}

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "basis_map_preserver", "input_interval": "(-inf, 0]", "output_interval": "(-inf, 0]", "statement": "(-inf, 0] -> (-inf, 0] through R_s, s = 3, and U_g" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map_theorem, SHA-256 350c6a330d73a4dc06505ded3b022085af3491ad0dfa1d0339704d5f9bf75e44, 2,618 bytes (1,071 compressed); download JSON

Cited by

• n-8229dbdd6a4b5cd9 (basis_map_preserver)

References

1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741