Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

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

Premises

Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
sequence: Every row P_n, n >= 0, 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-63e520ff3a0ab2b8 · 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\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 7156d8f92b3e166fce7cf0409abc93f4235e7233abd654e678022d6eff7c4967, 15,418 bytes (3,529 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-66e2c3448245b0b7 (hadamard_kernel)
n-66e2c3448245b0b7 (real_rooted)
n-66e2c3448245b0b7 (one_signed_coefficients)