Proof step e-452beb3fe1708cbf

basis_map A013611 → concludes about basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck … JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A013611",
  "operation": "basis_map"
}

Premises

• Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1… (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" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 1ff64610911cc64d5b94bb7c7ceb3559484cda28e51a5ba7943c058f5f8f77e9, 13,986 bytes (2,626 compressed); download JSON

Cited by

• n-146d64c3583c8509 (real_rooted)
• n-146d64c3583c8509 (one_signed_coefficients)
• n-146d64c3583c8509 (hadamard_kernel)
• n-146d64c3583c8509 (interlacing)