Proof step e-e31dcd4b09a7ff12

preserver_image affine, shift -1 → concludes about Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order). 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": -1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"0\"}"
}

Premises

• T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set. (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": 1, "region": { "lower": { "closed": true, "value": "0" } } }
• { "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 d96f79c9a9dbedd0a250792853d426c67def1cd1938e2433679ed90b1f1125ee, 15,075 bytes (3,186 compressed); download JSON

Cited by

• n-3de13f8c869a1c34 (one_signed_coefficients)
• n-3de13f8c869a1c34 (interlacing)
• n-3de13f8c869a1c34 (real_rooted)