A368539 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}.

1, 54, 761, 5284

Offset: 1

Author

Sela Fried, Dec 29 2023

Keywords

• nonn
• hard
• more

Comments

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

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

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

Examples

			                                                     [1 3 4]
For n = 3, the sum of the elements of A^2, where A = [2 6 8], is 761.
                                                     [5 7 9]
		

Links

• Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
• Hugo Pfoertner, Examples of solutions found by simulated annealing, for n=2-14. Jan 25, 2024.

Crossrefs

Cf. A085000, A350566, A369396.