Proof step e-556b2ae7482db11d

preserver_image affine, shift 0 → concludes about basis_map_A007318 of 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 >= … JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

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)
  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": { "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 f66efae47a2c9c94c0db1e55596480eaa13355f5a97a446563629bf7088f5b90, 15,439 bytes (3,709 compressed); download JSON

Cited by

• n-83e32d0bc1c10d45 (one_signed_coefficients)
• n-83e32d0bc1c10d45 (hadamard_kernel)
• n-83e32d0bc1c10d45 (interlacing)
• n-83e32d0bc1c10d45 (real_rooted)