Proof step e-33c9a461744a57dc

obreschkoff_combination → concludes about obreschkoff_combination 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

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

Operation

Obreschkoff combination Pₙ + Pₙ₊₁

{
  "operation": "obreschkoff_combination"
}

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 >= 2, is zero or real-rooted with all roots in (-inf, +inf)
• 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: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3

Claims

• { "claim": "real_rooted", "from_index": 2, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: obreschkoff_combination, SHA-256 13112f92adb1b0f2fed3b6d13a1adb1d6a4299f3f17c7bbe115ce9504f6f1530, 19,708 bytes (3,887 compressed); download JSON

Cited by

• n-ffeb6f77baa8bcee (real_rooted)

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.