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Maximal sum of elements of A^2 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.
2

%I #30 Feb 03 2024 10:14:39

%S 1,54,761,5284

%N Maximal sum of elements of A^2 where A is a square matrix of size n whose elements are a permutation of {1, 2, ..., n^2}.

%C The next terms are at least (and probably equal to) 5284, 24303, 85352 and 248045.

%C The lower bounds for the terms a(4)-a(7) are confirmed. a(8) >= 626610, a(9) >= 1421271, a(10) >= 2959798, a(11) >= 5750977. - _Hugo Pfoertner_, Jan 21 2024

%C In addition to the conditions (a)-(d) described in para 2.2 of Fried and Mansour (2023), conjecturally optimal matrices found using simulated annealing have the following additional property: If, using simultaneous row and column rearrangement, the matrix is brought into a form in which the terms of the main diagonal are sorted in ascending order, then every single row and every single column is monotonically increasing. See the linked file for examples from n=2 to n=14. - _Hugo Pfoertner_, Jan 25 2024

%H Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2308.00348">On the maximal sum of the entries of a matrix power</a>, arXiv:2308.00348 [math.CO], 2023.

%H Hugo Pfoertner, <a href="/A368539/a368539.txt">Examples of solutions found by simulated annealing</a>, for n=2-14. Jan 25, 2024.

%e [1 3 4]

%e For n = 3, the sum of the elements of A^2, where A = [2 6 8], is 761.

%e [5 7 9]

%Y Cf. A085000, A350566, A369396.

%K nonn,hard,more

%O 1,2

%A _Sela Fried_, Dec 29 2023