Proof step

Constant-current basis map

Basis map theorem for the sequence's own rule

Concludes: Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator de…
Premises: none: a direct theorem check
Theorem: [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ?
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-a5c879407de7a175 · JSON

Operation

Basis map theorem for the sequence's own rule

{
  "operation": "basis_map_theorem"
}

Claims

{
  "claim": "basis_map_preserver",
  "input_interval": "(-inf, 0]",
  "output_interval": "(-inf, -1]",
  "statement": "every PF input has a real-rooted image with every root <= -c/b = -1"
}

{
  "claim": "basis_map_preserver",
  "input_interval": "(-inf, 0]",
  "output_interval": "(-inf, -1]",
  "statement": "every PF input has a PF image (zero allowed)"
}

Certificate

basis_map_theorem, SHA-256 a9026baa767e8e596fd760588c0511344c06442c2db6d4da63cbef9f4ebfe88d, 3,201 bytes (1,068 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-72cbbb2f152c8f9c (basis_map_preserver)

References

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