Proof step e-3bcf5053bbc59f78

basis_map_theorem → concludes about Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2). 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, -1/2]", "statement": "every PF input has a real-rooted image with every root <= -c/b = -1/2" }
• { "claim": "basis_map_preserver", "input_interval": "(-inf, 0]", "output_interval": "(-inf, -1/2]", "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 92c7b0bd6a91258079ee952f8ba3794f2caa6db78a2ba3737f13e3cc3d076da3, 3,103 bytes (1,069 compressed); download JSON

Cited by

• n-7c641d81e21514ce (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