Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

shift_x_plus_one of Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Premises

Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-f6d6bfdc5b715676 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": false,
      "value": "-2"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 4cc22130690e3856be3c8ac9b5d10e3d8c16754f7ba3b64afbfc2f5cc58dcf95, 17,269 bytes (3,391 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-6c1b3ca24c1db31c (real_rooted)
n-6c1b3ca24c1db31c (one_signed_coefficients)
n-6c1b3ca24c1db31c (hadamard_kernel)
n-6c1b3ca24c1db31c (interlacing)