Proof step e-e5f3b61b305da328

preserver_image mobius, shift 0 → concludes about Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route fTransform RealRooted.splits_fPolynomial

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}}"
}

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)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 d33efe518ffa4da9037567a70fd5b4bde87d46da497d576ebad6fa9a2136a47a, 15,283 bytes (3,494 compressed); download JSON

Cited by

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