A369831 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the integers 1 to n.

1, 1, 1, 6, 23, 120, 720, 5040, 40320, 362880

Offset: 0

Author

Stefano Spezia, Feb 03 2024

Keywords

• nonn
• hard
• more

Links

• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A000142, A351019, A351020.

Cf. A369830, A369832, A369833, A369834, A369835.

Programs

Mathematica

a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Range[n]],i]]], {i, n!}]]; Join[{1}, Array[a,9]]

Python

from itertools import permutations
from sympy import Matrix
def A369831(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(range(1,n+1))}) # Chai Wah Wu, Feb 12 2024

Formula

a(n) <= A000142(n).

Conjectured e.g.f.: 1/(1 - x) - x^2/2 - x^4/24.