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A097694 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n. 4
36000, 50736, 69828, 94092, 124699, 162604, 208697, 264094, 329983, 407624, 498349, 603562 (list; graph; refs; listen; history; text; internal format)
OFFSET

15,1

LINKS

Table of n, a(n) for n=15..26.

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.

EXAMPLE

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.

Sequence in context: A236201 A034605 A031827 * A209956 A256365 A011766

Adjacent sequences:  A097691 A097692 A097693 * A097695 A097696 A097697

KEYWORD

more,nonn

AUTHOR

Hugo Pfoertner, Aug 24 2004

STATUS

approved

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Last modified May 20 20:15 EDT 2022. Contains 353876 sequences. (Running on oeis4.)