

A085000


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


30




OFFSET

1,2


COMMENTS

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
Improved lower bounds (private communication from Benjamin R. Buhrow, Dec 09 2019): a(8) >= 440970981670289, a(9) >= 346260899916111296.  Hugo Pfoertner, Jan 25 2021
Improved lower bound (private communication from Richard Gosiorovsky, Aug 18 2021): a(10) >= 356948996371054862392.  Hugo Pfoertner, Aug 24 2021


LINKS



EXAMPLE

The following 3 X 3 matrix is one of 36 whose determinant is 412 (there are also 36 whose determinant is 412):
9 3 5
4 8 1
2 6 7
Results from a specially adapted hillclimbing 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
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
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.


MATHEMATICA

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 *)


PROG

(PARI) vectomat(v)=my(n=sqrtint(#v)); matrix(n, n, i, j, v[n*(i1)+j])
a(n)=my(m, t, M); n*=n; for(k=0, (n1)!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


CROSSREFS

Cf. A088214, A088215, A088216, A088217, A088237, A180087 [upper bounds for a(n)], A180128, A221976, A301371, A301532, A301533, A309985, A325900, A350566.


KEYWORD

nonn,nice,hard,more


AUTHOR



EXTENSIONS

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
Entry edited by N. J. A. Sloane, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.


STATUS

approved



