Proof step e-548c39a476f786f5

preserver_image hadamard_multiplier, shift 1 → concludes about Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3. JSON

Theorem

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":1,\"coefficient_offset\":1}]}}"
}

Premises

• Triangle of Dyck paths counted by number of long interior inclines. (sequence)
  uses: Every row P_n, n >= 1, 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: preserver_image, SHA-256 c635e21ceaec08cd8cf1d94a21a365bac9f8450c477e8915408fff704a040168, 28,367 bytes (6,691 compressed); download JSON

Cited by

• n-047228a38014e6d4 (one_signed_coefficients)
• n-047228a38014e6d4 (hadamard_kernel)
• n-047228a38014e6d4 (real_rooted)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.