Proof step e-974c6818ff00e631

theorem_check liu_wang → concludes about Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k -… JSON

Theorem

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Operation

Theorem check of the node's own rule

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

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": false, "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 69e2c3ad9c5517015cb8ea6735cdc5f6686c69c3f699370cadc0f4b7cf11870d, 3,299 bytes (1,122 compressed); download JSON

Cited by

• n-1be5eefbe5bc79ec (real_rooted)
• n-1be5eefbe5bc79ec (one_signed_coefficients)
• n-1be5eefbe5bc79ec (hadamard_kernel)
• n-1be5eefbe5bc79ec (interlacing)

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003