A359617 a(n) is the maximal permanent of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

1, 1, 5, 54, 980, 26775, 1061841, 56647472, 4103545288, 367479636012

Offset: 0

Author

Stefano Spezia, Jan 07 2023

Keywords

• nonn
• hard
• more

Examples

			a(4) = 980:
  [   4,  3*i,  2*i,   i;
   -3*i,    4,  3*i, 2*i;
   -2*i, -3*i,    4, 3*i;
     -i, -2*i, -3*i,   4 ]
		

Links

• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A351020, A359560, A359562.

Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal).

Programs

Mathematica

a={1}; For[n=1, n<=7, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Permanent[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]])>mx, mx=t]]]; AppendTo[a, mx]]; a

Python

from itertools import permutations
from sympy import Matrix, I
def A359617(n): return max(Matrix(n,n,[(d[i-j] if i>j else -d[j-i]) if i!=j else d[0]*I for i in range(n) for j in range(n)]).per()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) if n else 1 # Chai Wah Wu, Jan 25 2023