Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, 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 read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1…
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-0e6aaa04faba66df · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

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

Certificate

preserver_image, SHA-256 cbeae48b971544b3c105910f5ebdf9cec539ec08267f9d2d1efb2b437e8cf98c, 14,282 bytes (3,410 compressed); download

polytool replayed the certificate when the step was written.

Cited by

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