Proof step e-928936169e50a847

basis_map_theorem → concludes about Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. JSON

Theorem

Constant-current basis map [Bra15, Thm 8.5] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_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": "every PF input has a real-rooted image with every root <= -c/b = 0" }
• { "claim": "basis_map_preserver", "input_interval": "(-inf, 0]", "output_interval": "(-inf, 0]", "statement": "every PF input has a PF image (zero allowed)" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map_theorem, SHA-256 7f03b85b72fca337649e33901479c7c042a6f78f28803ce2963955b185ed2a54, 3,138 bytes (1,038 compressed); download JSON

Cited by

• n-e189038e95d51606 (basis_map_preserver)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10