Proof step e-4c46966ead0b055f

gamma_polynomial scale 1/4, shift 0 → concludes about Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2. JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 0,
  "operation": "gamma_polynomial",
  "scale": "1/4"
}

Premises

• type_b_eulerian (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: gamma_polynomial, SHA-256 5b3e4033348c4d712d4422010d63f6ad0e8ad6805080080beb4af51b62151a0e, 10,103 bytes (2,280 compressed); download JSON

Cited by

• n-5e3f8ad0b0a35d3e (real_rooted)
• n-5e3f8ad0b0a35d3e (one_signed_coefficients)
• n-5e3f8ad0b0a35d3e (hadamard_kernel)