Proof step

Iterated real-rootedness preservers

Theorem check of the sequence's own rule iterated preserver

Concludes: Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.
Premises: none: a direct theorem check
Theorem: [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ?
Lean: RealRooted RealRooted.IsPFPolynomial.generalStep_of_natDegree_le, RealRooted.EulerBidiagonal.generalStep, RealRooted.EulerBidiagonal.comparisonDefect, RealRooted.EulerBidiagonal.generalRowsDep_spec
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-ad824f97497bc135 · JSON

Operation

Theorem check of the sequence's own rule

{
  "operation": "theorem_check",
  "theorem": "iterated_preserver"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

theorem_check, SHA-256 18730f2f125ef0debb08e42fed62a40169684ad5fc886cf4274a790e9f636b78, 8,940 bytes (2,118 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.