Real-rooted atlas

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 binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).
Premises
none: a direct theorem check
Theorem
[SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ?
Checked
polytool replayed · Lean not attempted
Technical details
Step id
e-37ac46811d9937f6 · JSON
Operation
Theorem check of the sequence's own rule
{
  "operation": "theorem_check",
  "theorem": "iterated_preserver"
}
Claims
{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}
Certificate
theorem_check, SHA-256 37abffe6d51a5af3c35ba77ae8ec4cf2896fa7c72979192b4c7830e96829bcf4, 3,800 bytes (1,224 compressed); download
polytool replayed the certificate when the step was written.
Cited by

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.