Proof step

Constant-current basis map

Basis map theorem for the sequence's own rule

Concludes: Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a…
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-bd3929b5a5b502dd · 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, 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)"
}

Certificate

basis_map_theorem, SHA-256 6e6e2a2eadec4be1c25ab6bbf4917c5f4a5bc895a27175f4bb9abfc4c2a936d7, 3,209 bytes (1,071 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-a5d9cd4cf05da467 (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