Real-rooted atlas

Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes
Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
Premises
  • eulerian
    sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
Theorem
polytool ?
Lean
RealRooted route fTransform RealRooted.splits_fPolynomial
Checked
polytool replayed · Lean not attempted
Technical details
Step id
e-8205223e631a4605 · JSON
Operation
Image under a real-rootedness preserver
{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"
}
Claims
{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}
Certificate
preserver_image, SHA-256 81c9195c22d5b4c81b691075bd163164b2dc6284c7ad9a12d1d32a67a1c89845, 10,892 bytes (2,708 compressed); download
polytool replayed the certificate when the step was written.
Cited by