Proof step

Ma–Wang interval theorem

Theorem check of the sequence's own rule ma wang interval

Concludes: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
Premises: none: a direct theorem check
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
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-598b560f6729035b · JSON

Operation

Theorem check of the sequence's own rule

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

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": false,
      "value": "-1"
    },
    "upper": {
      "closed": false,
      "value": "0"
    }
  }
}

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

Certificate

theorem_check, SHA-256 e50d200c2737ba7c5eb9bcd8ac24c976c0ae57cd968f25ea94719dd20bc2cb95, 3,974 bytes (1,307 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-d938c1937a93c4df (real_rooted)
n-d938c1937a93c4df (one_signed_coefficients)
n-d938c1937a93c4df (hadamard_kernel)
n-d938c1937a93c4df (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