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.

1, 1, 5, 18, 245, 2249, 57213, 947177, 50431724, 1282453618

Offset: 0

Author

Stefano Spezia, Jan 07 2023

Keywords

• nonn
• hard
• more

Examples

			a(4) = 245:
  [   1,  3*i,  2*i, 4*i;
   -3*i,    1,  3*i, 2*i;
   -2*i, -3*i,    1, 3*i;
   -4*i, -2*i, -3*i,   1 ]
		

Links

• Wikipedia, Toeplitz Matrix.

Crossrefs

Cf. A351019, A359560, A359562.

Cf. A359614 (minimal determinant), A359615 (maximal determinant), A359617 (maximal).

Programs

Mathematica

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]]]])
				

Python

from itertools import permutations
from sympy import Matrix, I
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