Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

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: Every row P_n, n >= 1, 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-0e8ec468368c6751 · 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": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 2f00f08575c7e9780cf2f266b4104d7ed628ea38dd0ec935f84c26a1db7f879c, 19,101 bytes (4,075 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-1cc5242f219c3841 (real_rooted)
n-1cc5242f219c3841 (one_signed_coefficients)
n-1cc5242f219c3841 (hadamard_kernel)
n-1cc5242f219c3841 (interlacing)

References

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