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A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.

%I A350566 #25 Aug 30 2025 10:12:44
%S A350566 1,1,14,947,161388,56558003,36757837732
%N A350566 a(n) is the maximum permanent of an n X n matrix using the integers 1 to n^2.
%C A350566 a(7) >= 38677620556961 corresponding to the matrix
%C A350566   14, 25, 39,  3, 45,  2, 42
%C A350566   32, 21, 10, 46,  5, 47,  8
%C A350566   31, 20,  9, 48,  1, 49,  6
%C A350566   44, 24, 18, 33, 13, 34, 15
%C A350566   22, 29, 35, 12, 36, 11, 37
%C A350566   16, 26, 38,  7, 43,  4, 40
%C A350566   23, 41, 30, 19, 27, 17, 28 . - _Robert Israel_, Mar 19 2025
%C A350566 a(7) >= 38677691168324 corresponding to the matrix
%C A350566    1,  4, 14, 25, 39, 42, 45
%C A350566    5,  6, 16, 26, 38, 40, 43
%C A350566   11, 12, 22, 29, 35, 36, 37
%C A350566   17, 19, 23, 41, 30, 28, 27
%C A350566   33, 34, 44, 24, 18, 15, 13
%C A350566   48, 46, 32, 21, 10,  8,  3
%C A350566   49, 47, 31, 20,  9,  7,  2. - _Pontus von Brömssen_, Mar 20 2025
%H A350566 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 A350566 a(2) = 14:
%e A350566   [2, 3;
%e A350566    4, 1]
%e A350566 .
%e A350566 a(3) = 947:
%e A350566   [3, 7, 6;
%e A350566    9, 4, 1;
%e A350566    2, 5, 8]
%e A350566 .
%e A350566 a(4) = 161388:
%e A350566   [ 2,  3, 16,  6;
%e A350566    11, 13,  4, 10;
%e A350566     8,  9,  5, 15;
%e A350566    14, 12,  1,  7]
%e A350566 .
%e A350566 a(5) = 56558003:
%e A350566   [10,  2, 19, 25,  3;
%e A350566    11,  5, 23, 20,  8;
%e A350566    21, 14, 12,  9, 15;
%e A350566    13, 24,  6,  1, 18;
%e A350566    16, 17,  7,  4, 22]
%e A350566 .
%e A350566 a(6) = 36757837732:
%e A350566   [32, 30,  3, 19, 23,  2;
%e A350566     1,  5, 34, 14, 11, 36;
%e A350566    17, 18, 15, 31, 22, 16;
%e A350566    29, 28,  7, 20, 24,  6;
%e A350566    26, 25, 10, 21, 27,  9;
%e A350566     4,  8, 35, 13, 12, 33]
%o A350566 (Python)
%o A350566 from itertools import permutations
%o A350566 from sympy import Matrix
%o A350566 def A350566(n): return 1 if n == 0 else max(Matrix(n,n,p).per() for p in permutations(range(1,n**2+1))) # _Chai Wah Wu_, Jan 21 2022
%Y A350566 Cf. A085000 (determinant), A350565 (minimum), A350858, A350859, A358487 (elements 0 to n^2-1).
%K A350566 nonn,hard,more,changed
%O A350566 0,3
%A A350566 _Hugo Pfoertner_ at the suggestion of _Stefano Spezia_, Jan 21 2022