%I #5 Mar 31 2012 10:29:05
%S 36000,50736,69828,94092,124699,162604,208697,264094,329983,407624,
%T 498349,603562
%N Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n.
%F For n<=18 an optimal choice and arrangement is of the following form det((n, 6, 2, n8), (3, n1, 4, n5), (n7, n6, n2, 0), (5, 1, n4, n3))= 3*n^462*n^3+543*n^22128*n+3120. For n>18 up to the limit investigated (n<=26) one choice maximizing the determinant is det((n, 6, 2, n8), (3, n1, 4, n5), (n7, n6, n3, 0), (5, 1, n2, n4))= 3*n^460*n^3+491*n^21821*n+2624. In both cases there are 575 other equivalent arrangements corresponding to permutations of rows and columns.
%e a(18)=94092 because no 4 X 4 matrix b(j,k) with distinct elements 0<=b(j,k)<=18,j=1..4,k=1..4 can be built that has a larger determinant than
%e det((18,6,2,10),(3,17,4,13),(11,12,16,0),(5,1,14,15)=94092.
%Y Other maximal 4 X 4 determinants: Cf. A097696: 4 X 4 matrix filled with consecutive integers, A097695: 4 X 4 matrix filled with integers from n...n, A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices, A085000: n X n matrix filled with consecutive integers 1...n^2.
%K more,nonn
%O 15,1
%A _Hugo Pfoertner_, Aug 24 2004
