A097694 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n.

36000, 50736, 69828, 94092, 124699, 162604, 208697, 264094, 329983, 407624, 498349, 603562

Offset: 15

Author

Hugo Pfoertner, Aug 24 2004

Keywords

• more
• nonn

Examples

			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
det((18,6,2,10),(3,17,4,13),(11,12,16,0),(5,1,14,15))=94092.
		

Crossrefs

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.

Formula

For n<=18 an optimal choice and arrangement is of the following form det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-2, 0), (5, 1, n-4, n-3))= 3*n^4-62*n^3+543*n^2-2128*n+3120. For n>18 up to the limit investigated (n<=26) one choice maximizing the determinant is det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-3, 0), (5, 1, n-2, n-4))= 3*n^4-60*n^3+491*n^2-1821*n+2624. In both cases there are 575 other equivalent arrangements corresponding to permutations of rows and columns.