Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

Premises

Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-dabac2fe17670051 · 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": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 638e8a02c5aeb16237a8705bc531e9c46d74ffcc5d5968dad7908b004acd4396, 10,757 bytes (2,839 compressed); download

polytool replayed the certificate when the step was written.

Cited by