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A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.

%I A350565 #21 Aug 31 2025 03:09:28
%S A350565 1,1,10,436,52197,13300936,6192060119
%N A350565 a(n) is the minimum permanent of an n X n matrix using the integers 1 to n^2.
%C A350565 a(7) <= 4755379618016 from the matrix
%C A350565   [ 1,  2,  3,  4,  5,  6,  7;
%C A350565     8, 14, 19, 23, 29, 33, 36;
%C A350565     9, 15, 21, 26, 32, 37, 41;
%C A350565    10, 16, 22, 28, 35, 40, 44;
%C A350565    11, 17, 24, 30, 38, 43, 46;
%C A350565    12, 18, 25, 31, 39, 45, 48;
%C A350565    13, 20, 27, 34, 42, 47, 49]. - _Pontus von Brömssen_, Aug 30 2025
%H A350565 Carl-Erik Fröberg, <a href="https://doi.org/10.1007/BF01941124">On a combinatorial problem related to permanents</a>, BIT 28 (1988), No. 3, 406-411.
%e A350565 a(2) = 10:
%e A350565   [1, 3;
%e A350565    2, 4]
%e A350565 .
%e A350565 a(3) = 436:
%e A350565   [1, 3, 2;
%e A350565    4, 8, 6;
%e A350565    5, 9, 7]
%e A350565 .
%e A350565 a(4) = 52197:
%e A350565   [1,  2,  4,  3;
%e A350565    6,  9, 15, 12;
%e A350565    5,  8, 13, 11;
%e A350565    7, 10, 16, 14]
%e A350565 .
%e A350565 a(5) = 13300936:
%e A350565   [16,  8, 24, 21, 12;
%e A350565    18,  9, 25, 23, 13;
%e A350565     3,  1,  5,  4,  2;
%e A350565    14,  6, 20, 17, 10;
%e A350565    15,  7, 22, 19, 11]
%e A350565 .
%e A350565 a(6) = 6192060119:
%e A350565   [36, 35, 33, 31, 27,  6;
%e A350565    11, 10,  9,  8,  7,  1;
%e A350565    34, 32, 30, 28, 25,  5;
%e A350565    22, 21, 19, 18, 16,  3;
%e A350565    29, 26, 24, 23, 20,  4;
%e A350565    17, 15, 14, 13, 12,  2]
%o A350565 (Python)
%o A350565 from itertools import permutations
%o A350565 from sympy import Matrix
%o A350565 def A350565(n): return 1 if n == 0 else min(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # _Chai Wah Wu_, Jan 21 2022
%Y A350565 Cf. A085000, A350566 (maximum), A350858, A350859, A358486 (elements 0 to n^2-1).
%K A350565 nonn,hard,more,changed
%O A350565 0,3
%A A350565 _Stefano Spezia_ and _Hugo Pfoertner_, Jan 20 2022