Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

Premises

Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
sequence: Every row P_n, n >= 0, 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-b1f2d9eaf84abe9d · 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": {
    "lower": {
      "closed": false,
      "value": "-2"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

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

Certificate

preserver_image, SHA-256 1083c22d79f0ec2b1c07bed2c0f7c235eac4afa0c527022483f1c92b0ce026df, 20,372 bytes (3,816 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-07af2e6e3c9f0c77 (hadamard_kernel)
n-07af2e6e3c9f0c77 (interlacing)
n-07af2e6e3c9f0c77 (real_rooted)
n-07af2e6e3c9f0c77 (one_signed_coefficients)