Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

Premises

T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-98ed2204a3519c03 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 97f5493573503bf56ab3bd3935e75017c99d5a7cd43986e71275d58e4ad12a9c, 10,690 bytes (2,715 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-8ab17db79dad1880 (real_rooted)
n-8ab17db79dad1880 (one_signed_coefficients)
n-8ab17db79dad1880 (hadamard_kernel)