A350939 - OEIS

%I #22 Feb 12 2024 10:41:29

%S 1,2,19,496,29609,3009106,498206489

%N Minimal permanent of an n X n Toeplitz matrix using the first 2*n - 1 prime numbers.

%C For n X n Hankel matrices the same minimal permanents appear.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A350939%2B40.sage">A350939+40.sage</a>

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

%e a(2) = 19:

%e     2    3

%e     5    2

%e a(3) = 496:

%e     2    3    7

%e     5    2    3

%e    11    5    2

%t a[n_] := Min[Table[Permanent[HankelMatrix[Join[Drop[per = Part[Permutations[Prime[Range[2 n - 1]]], i], n], {Part[per, n]}], Join[{Part[per, n]}, Drop[per, - n]]]], {i, (2 n - 1) !}]]; Join[{1}, Array[a, 5]] (* _Stefano Spezia_, Feb 06 2024 *)

%o (Python)

%o from itertools import permutations

%o from sympy import Matrix, prime

%o def A350939(n): return 1 if n == 0 else min(Matrix([p[n-1-i:2*n-1-i] for i in range(n)]).per() for p in permutations(prime(i) for i in range(1,2*n))) # _Chai Wah Wu_, Jan 27 2022

%o (PARI) a(n) = my(v=[1..2*n-1], m=+oo, d); forperm(v, p, d = matpermanent(matrix(n, n, i, j, prime(p[i+j-1]))); if (d<m, m = d)); m; \\ _Michel Marcus_, Feb 08 2024

%Y Cf. A318173, A350932, A350940 (maximal), A290302, A350937, A369952.

%K nonn,hard,more

%O 0,2

%A _Stefano Spezia_, Jan 26 2022

%E a(5) from _Alois P. Heinz_, Jan 26 2022

%E a(6) from _Lucas A. Brown_, Sep 05 2022