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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.

%I A359617 #21 Jan 25 2023 22:07:22
%S A359617 1,1,5,54,980,26775,1061841,56647472,4103545288,367479636012
%N 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.
%H A359617 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A359617 a(4) = 980:
%e A359617   [   4,  3*i,  2*i,   i;
%e A359617    -3*i,    4,  3*i, 2*i;
%e A359617    -2*i, -3*i,    4, 3*i;
%e A359617      -i, -2*i, -3*i,   4 ]
%t A359617 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
%o A359617 (Python)
%o A359617 from itertools import permutations
%o A359617 from sympy import Matrix, I
%o A359617 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
%Y A359617 Cf. A351020, A359560, A359562.
%Y A359617 Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359616 (minimal).
%K A359617 nonn,hard,more
%O A359617 0,3
%A A359617 _Stefano Spezia_, Jan 07 2023