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 A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2. 22

%I

%S 1,10,412,40800,6839492,1865999570,762150368499

%N Maximal determinant of an n X n matrix using the integers 1 to n^2.

%C Bounds for the next terms and the corresponding matrices are given by O. Gasper, H. Pfoertner and M. Sigg: 440960274696935<=a(8)<441077015225642, 346254605664223620<=a(9)<346335386150480625, 356944784622927045792<=a(10)<357017114947987625629. a(n)<sqrt(3*((n^5+n^4+n^3+n^2)/12)^n*(n^2+1)/(n+1)). - _Hugo Pfoertner_, Aug 15 2010

%H Ortwin Gasper, Hugo Pfoertner and Markus Sigg, <a href="http://www.emis.de/journals/JIPAM/article1119.html">An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum</a>, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a085000.txt">Maximal determinant of matrix with elements 1..n.</a> FORTRAN program.

%H Markus Sigg, <a href="https://arxiv.org/abs/1804.02897">Gasper's determinant theorem, revisited</a>, arXiv:1804.02897 [math.CO], 2018.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareMatrix.html">Square Matrix</a>

%H <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>

%e The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is -412):

%e 9 3 5

%e 4 8 1

%e 2 6 7

%e Results from a specially-adapted hill-climbing algorithm strongly suggest that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, a(n) is approximately 0.44 * n^(2.06*n), suggesting that a(7) is close to 6.8 * 10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 2003

%e a(5) confirmed by _Robert Israel_ and _Hugo Pfoertner_. A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 20 5) (10 13 2 22 19) (16 1 18 8 21) ). - _Hugo Pfoertner_, Sep 23 2003

%e a(6) found with FORTRAN program given at Pfoertner link. A corresponding matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ). - _Hugo Pfoertner_, Sep 23 2003

%e a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although no proof for the optimality of a(7) is available, the results of an extensive computational search make the existence of a better solution extremely unlikely. A total of approximately 15 CPU years on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers was used for this result.

%t Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], {k, n2!}]; {dMax, mMax} (* _T. D. Noe_ *)

%o (PARI) vectomat(v)=my(n=sqrtint(#v));matrix(n,n,i,j,v[n*(i-1)+j])

%o a(n)=my(m,t,M); n*=n; for(k=0,(n-1)!-1, t=matdet(M=vectomat(numtoperm(n,k))); if(abs(t)>m, m=abs(t); print(t" "M)));m \\ _Charles R Greathouse IV_, Sep 13 2013

%Y Cf. A088214, A088215, A088216, A088217, A088237, A180087 [upper bounds for a(n)], A180128, A221976, A301371, A301532, A301533, A309985, A325900.

%K nonn,nice,more,hard

%O 1,2

%A _Robert G. Wilson v_, Jun 16 2003

%E a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003

%E a(6) from _Hugo Pfoertner_, Sep 23 2003

%E Entry edited by _N. J. A. Sloane_, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.

%E a(7) from _Hugo Pfoertner_, Jan 22 2008

%E Comment updated and reference added by _Hugo Pfoertner_, Aug 15 2010

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Last modified September 29 14:13 EDT 2020. Contains 337431 sequences. (Running on oeis4.)