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Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.
9

%I #20 Oct 11 2022 00:54:50

%S 1,1,-3,-12,-100,-1749,-47600,-800681,-39453535,-1351201968,

%T -66984136299,-2938096403400,-235011452211680

%N Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A350953%2B4%2BA356865.py">A350953+4+A356865.py</a>

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

%e a(3) = -12:

%e 2 3 1

%e 3 2 3

%e 1 3 2

%e a(4) = -100:

%e 3 4 1 2

%e 4 3 4 1

%e 1 4 3 4

%e 2 1 4 3

%e a(5) = -1749:

%e 5 4 1 3 2

%e 4 5 4 1 3

%e 1 4 5 4 1

%e 3 1 4 5 4

%e 2 3 1 4 5

%o (Python)

%o from itertools import permutations

%o from sympy import Matrix

%o def A350953(n): return min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # _Chai Wah Wu_, Jan 27 2022

%Y Cf. A307887, A350930, A350954 (maximal), A356865 (minimal nonzero absolute value).

%K sign,hard,more

%O 0,3

%A _Stefano Spezia_, Jan 27 2022

%E a(9) from _Alois P. Heinz_, Jan 27 2022

%E a(10)-a(12) from _Lucas A. Brown_, Sep 01 2022