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A359616 a(n) is the minimal 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 A359616 #23 Jan 25 2023 21:10:57
%S A359616 1,1,5,18,245,2249,57213,947177,50431724,1282453618
%N A359616 a(n) is the minimal 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 A359616 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A359616 a(4) = 245:
%e A359616   [   1,  3*i,  2*i, 4*i;
%e A359616    -3*i,    1,  3*i, 2*i;
%e A359616    -2*i, -3*i,    1, 3*i;
%e A359616    -4*i, -2*i, -3*i,   1 ]
%t A359616 a={1}; For[n=1, n<=7, n++, mn=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]]]])<mn, mn=t]]]; AppendTo[a, mn]]; a
%o A359616 (Python)
%o A359616 from itertools import permutations
%o A359616 from sympy import Matrix, I
%o A359616 def A359616(n): return min(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 A359616 Cf. A351019, A359560, A359562.
%Y A359616 Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359617 (maximal).
%K A359616 nonn,hard,more
%O A359616 0,3
%A A359616 _Stefano Spezia_, Jan 07 2023