Proof step e-ad13a5d17ca116d8

preserver_image reversal, shift 1 → concludes about Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Operation

Image under a real-rootedness preserver

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

Premises

• Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "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 4dc5c386685816e48283447cb1f87d6cee9b7690e717cc363e9b46c8b0f30a0d, 13,976 bytes (2,932 compressed); download JSON

Cited by

• n-e65f982fa1b436c5 (real_rooted)
• n-e65f982fa1b436c5 (one_signed_coefficients)
• n-e65f982fa1b436c5 (hadamard_kernel)