Proof step e-a4c8bd871ea20676

basis_map_theorem → concludes about Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals. 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 0e577a35c4229481ab3fe79d3236c1e70cd92b7908d45972ed48dc273f2bd2d4, 3,123 bytes (1,024 compressed); download JSON

Cited by

• n-c8b3f39d03de798f (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