Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

shift_x_plus_one of 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

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.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, -1)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-601c9dcf57fce44e · 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": "-3"
    },
    "upper": {
      "closed": false,
      "value": "-2"
    }
  }
}

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

Certificate

preserver_image, SHA-256 7c9ed3dd5786a6a1c301a86e89cefb3321e221735a668bd37e33b986f62a9f90, 20,780 bytes (4,169 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-fb13dff3dfb4d666 (real_rooted)
n-fb13dff3dfb4d666 (one_signed_coefficients)
n-fb13dff3dfb4d666 (hadamard_kernel)
n-fb13dff3dfb4d666 (interlacing)