Proof step e-2ff0affdc4d5d50d

basis_map A049020 → concludes about Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished. JSON

Theorem

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A049020",
  "operation": "basis_map"
}

Premises

• a(n) = 1 if n is of the form m(m+3)/2, otherwise 0. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-1" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 f8e3649361ce95670af2a508cf6a563f694912f2bde832121c477732c2966d96, 7,541 bytes (1,498 compressed); download JSON

Cited by

• n-c15181c2ebe13c1d (real_rooted)
• n-c15181c2ebe13c1d (one_signed_coefficients)
• n-c15181c2ebe13c1d (hadamard_kernel)

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741