A359615 a(n) is the maximal determinant 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, 3, 9, 512, 9195, 242931, 7459494, 524426191, 17012915860, 773407040859

Offset: 0

Author

Stefano Spezia, Jan 07 2023

Keywords

• nonn
• hard
• more

Examples

			a(4) = 512:
  [   1,  4*i,  2*i, 3*i;
   -4*i,    1,  4*i, 2*i;
   -2*i, -4*i,    1, 4*i;
   -3*i, -2*i, -4*i,   1 ]
		

Links

• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A350954, A359559, A359561.

Cf. A359614 (minimal), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

Mathematica

a={1}; For[n=1, n<=8, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Det[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 A359615(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)]).det()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) # Chai Wah Wu, Jan 25 2023