A359615 - OEIS

%I #23 Jan 25 2023 20:50:17

%S 1,1,3,9,512,9195,242931,7459494,524426191,17012915860,773407040859

%N 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.

%e a(4) = 512:

%e   [   1,  4*i,  2*i, 3*i;

%e    -4*i,    1,  4*i, 2*i;

%e    -2*i, -4*i,    1, 4*i;

%e    -3*i, -2*i, -4*i,   1 ]

%t 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

%o (Python)

%o from itertools import permutations

%o from sympy import Matrix, I

%o 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

%Y Cf. A350954, A359559, A359561.

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

%K nonn,hard,more

%O 0,3

%A _Stefano Spezia_, Jan 07 2023