Proof step `e-ade83b202f7a862b`

theorem_check ma_wang_interval → concludes about Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n. JSON

Theorem

Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Lean (RealRooted): RealRooted.affineEndpointStep_closed, RealRooted.affineEndpointStep_roots_mem_Icc_iff, RealRooted.affineEndpointStep_rootMultiplicity_zero, RealRooted.affineEndpointStep_rootMultiplicity_neg_one, RealRooted.MaWang.endpointStep, RealRooted.MaWang.affineEndpointSequence, RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff

Operation

Theorem check of the node's own rule

{
  "operation": "theorem_check",
  "theorem": "ma_wang_interval"
}

Premises

None: this step is a direct theorem check.

Claims

{ "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-1" }, "upper": { "closed": true, "value": "0" } } }
{ "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "previous_interlaces_next" }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: theorem_check, SHA-256 39526b1b9d0713372293d19c4573d9b166bcfa53ba394addc6f0b68730fc11aa, 4,016 bytes (1,329 compressed); download JSON

Cited by

n-eda3c556a6b4857f (real_rooted)
n-eda3c556a6b4857f (one_signed_coefficients)
n-eda3c556a6b4857f (hadamard_kernel)
n-eda3c556a6b4857f (interlacing)

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741

Proof step e-ade83b202f7a862b