Proof step e-bcbfbcbb753a93dd

preserver_image affine, shift 0 → concludes about Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished. 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 of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n. (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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 01467da03f8d861a9fde153ef8f9b26beec1e59a1b4ce7070fa7ebc47d1b8237, 6,916 bytes (2,063 compressed); download JSON

Cited by

• n-c15181c2ebe13c1d (real_rooted)
• n-c15181c2ebe13c1d (one_signed_coefficients)
• n-c15181c2ebe13c1d (hadamard_kernel)