Proof step

Laguerre’s theorem on differential operators

Image under a real-rootedness preserver derivative polynomial, shift 0

Concludes

derivative of Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).

Premises

Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14] polytool ?

Lean

RealRooted RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-f8a4d338a086df20 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 ffb76bdbea50c688923ebc6f9f14c0299a4cc06c9e6dddfc6476b9be634b65c2, 11,172 bytes (3,073 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-671db268e7aa46c0 (real_rooted)
n-671db268e7aa46c0 (one_signed_coefficients)
n-671db268e7aa46c0 (hadamard_kernel)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.