Proof step `e-63c86030c8ae0f8e`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_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. JSON

Theorem

Laguerre’s theorem on differential operators [SP14] polytool ?

Lean (RealRooted): RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"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": "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 1957f73e38e6542438bd003fe7a7ba349f4670d99142dd2fb537f199da31d25e, 20,247 bytes (4,306 compressed); download JSON

Cited by

n-0aa2174fd7ab3229 (interlacing)
n-0aa2174fd7ab3229 (real_rooted)
n-0aa2174fd7ab3229 (one_signed_coefficients)
n-0aa2174fd7ab3229 (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.

Proof step e-63c86030c8ae0f8e