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%I #3 Mar 31 2012 10:29:05
%S 10324,19920,35791,60122,95610,145362
%N Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct integers chosen from the range -n...n.
%C The formula for a(12) and a(13) gives lower bounds for the next terms a(14)>=212802, a(15)>=301770.
%F Optimal choices and arrangements: n=8 see example, n=9, 10, 11: det((n, n-1, 7-n, n-5), (5-n, n-4, 1-n, 4-n), (n-2, 2-n, 3-n, 6-n), (n-6, n-7, n-3, -n))= 16*n^4-224*n^3+1334*n^2-3795*n+4341. n=12, 13: det((n, n-5, n-3, 6-n), (n-6, -n, 5-n, 3-n), (7-n, n-1, 4-n, 2-n), (n-2, n-7, 1-n, n-4))= 2*(8*n^4-112*n^3+670*n^2-1947*n+2325). For each n there are (4!)^2=576 equivalent arrangements corresponding to permutations of rows and columns.
%e a(8)=10324 because no larger determinant of a 4 X 4 integer matrix b(j,k) with distinct elements -8<=b(j,k)<=8,j=1..4,k=1..4 can be built than
%e det((8,4,3,2),(-1,7,-4,-6),(5,-5,-7,-2),(1,-3,6,-8))=10324.
%Y Other maximal 4 X 4 determinants: Cf. A097694: 4 X 4 matrix filled with integers from 0...n, A097696: 4 X 4 matrix filled with consecutive integers. A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices.
%K more,nonn
%O 8,1
%A _Hugo Pfoertner_, Aug 25 2004
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