%I
%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, n1, 7n, n5), (5n, n4, 1n, 4n), (n2, 2n, 3n, 6n), (n6, n7, n3, n))= 16*n^4224*n^3+1334*n^23795*n+4341. n=12, 13: det((n, n5, n3, 6n), (n6, n, 5n, 3n), (7n, n1, 4n, 2n), (n2, n7, 1n, n4))= 2*(8*n^4112*n^3+670*n^21947*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
