Proof step e-c002970bb7812a06

basis_map_theorem → concludes about Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}. 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 92df0d7c2d5ac61a89bbf2f1758cbd93f5015a979a954c8d963ac42b924fffe8, 3,057 bytes (1,036 compressed); download JSON

Cited by

• n-ba3bb18490b15e2e (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