Real-rooted atlas

Proof step

Veronese sections of Pólya frequency sequences

Veronese coefficient section r=2, j=1, shift 0

Concludes
veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 til…
Premises
Theorem
[ASW52; Edr53; AW24] polytool ?
Checked
polytool replayed · Lean not attempted
Technical details
Step id
e-8ca693cf2e4d644e · JSON
Operation
Veronese coefficient section
{
  "index_shift": 0,
  "modulus": 2,
  "operation": "veronese_section",
  "residue": 1
}
Claims
{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}
{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "previous_interlaces_next"
}
Certificate
veronese_section, SHA-256 7de65d3464ffff026faff4859d4d1e62b5f9e981ea806dd6c6d7bc9d41d45af0, 9,913 bytes (2,669 compressed); download
polytool replayed the certificate when the step was written.
Cited by

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3