Proof step e-5885e4696854d233

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents. 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 permutations of 1..n arranged in a circle with exactly k descents. (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": 1, "region": { "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 2, "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 216074aba97618879eb9ace0fa4e0f4fe2d9a926ebd6069627e6f5333fa33e8c, 16,856 bytes (3,776 compressed); download JSON

Cited by

• n-5a739e9e90502f2d (interlacing)
• n-5a739e9e90502f2d (real_rooted)
• n-5a739e9e90502f2d (one_signed_coefficients)
• n-5a739e9e90502f2d (hadamard_kernel)