Proof step e-404a8e815c57c4e9

preserver_image euler_polynomial, shift 0 → concludes about theta_plus_one_squared of 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

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Lean (RealRooted): RealRooted.isMultiplierSequence_eval_of_roots_nonpos, RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos, RealRooted.isMultiplierSequence_iff_preserves

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

• Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents. (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": 1, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 c1675632a88c1a33fcbf3ad7a5e619a640103864cbc9862dce28c3400e0e8576, 21,980 bytes (4,627 compressed); download JSON

Cited by

• n-73486cfabd571954 (real_rooted)
• n-73486cfabd571954 (one_signed_coefficients)
• n-73486cfabd571954 (hadamard_kernel)
• n-73486cfabd571954 (interlacing)

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.