Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

Premises

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.
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-6c3e82dc47b23e3d · 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 700562a4bb1ca0dbd418871d0b06c1d91d7ddff135fb4302bc27cce15f7d4b9f, 10,666 bytes (2,720 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-47fe395e9f7d5f86 (one_signed_coefficients)
n-47fe395e9f7d5f86 (hadamard_kernel)
n-47fe395e9f7d5f86 (real_rooted)