Proof step e-9baf8deff42ffdae

theorem_check liu_wang_derivative_lag → concludes about 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

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Lean (RealRooted): route halfGrowth RealRooted.derivLag_strictInterl_of_nonnegCoeffs, RealRooted.derivLag_strictInterl_and_noCommonRoot, RealRooted.derivLag_eval_zero_pos

Operation

Theorem check of the node's own rule

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

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "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 447d3e61ec8dd9e298d858fbb4b916ce19bdd0e612e5095f4b4d11a30fadbca9, 6,452 bytes (1,827 compressed); download JSON

Cited by

• n-3ffff24fcb4f40a0 (real_rooted)
• n-3ffff24fcb4f40a0 (one_signed_coefficients)
• n-3ffff24fcb4f40a0 (hadamard_kernel)
• n-3ffff24fcb4f40a0 (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