A325900 -id:A325900 - cp's OEIS frontend

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

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

Offset: 1

Robert G. Wilson v, Jun 16 2003

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

			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 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
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
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.

Sela Fried and Toufik Mansour, On the maximal sum of the entries of a matrix power, arXiv:2308.00348 [math.CO], 2023.
Ortwin Gasper, Hugo Pfoertner and Markus Sigg, An Upper Bound for the Determinant of a Matrix with given Entry Sum and Square Sum, JIPAM, Journal of Inequalities in Pure and Applied Mathematics, Volume 10, Issue 3, Article 63, 2008.
Hugo Pfoertner, Maximal determinant of matrix with elements 1..n. FORTRAN program.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Square Matrix
Index entries for sequences related to maximal determinants

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

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 *)

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

a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
a(6) from Hugo Pfoertner, Sep 23 2003
Entry edited by N. J. A. Sloane, Nov 22 2006, to remove some erroneous entries. Further edits Nov 25 2006.
a(7) from Hugo Pfoertner, Jan 22 2008

A088216 Smallest nonnegative number not expressible as determinant of an n X n matrix with elements 1..n^2.

Original entry on oeis.org

0, 324, 38831, 6773999, 1859163031

Offset: 2

Hugo Pfoertner, Sep 23 2003

			a(2)=0 because the 2 X 2 determinant of a matrix with entries that are permutations of 1,2,3,4 can only assume the values +-2,+-5,+-10.

Hugo Pfoertner, Find all possible values of det(A) for matrix with elements 1..n^2. FORTRAN program.

a(3)=A088214(1), a(4)=A088237(1), a(5)=A088238(1), a(6)=A325900(1).

Cf. A085000, A088215, A088217, A322576.

a(6) from Hugo Pfoertner, Sep 07 2019

A088217 Number of distinct values that can be assumed by the determinant of an n X n matrix whose entries are all permutations of the numbers 1..n^2.

Original entry on oeis.org

1, 6, 777, 79455, 13602389, 3722956267

Offset: 1

Hugo Pfoertner, Sep 23 2003

a(5) = 1 + 2*(A085000(5) - (number of terms of A088238)).

			a(2)=6 because the determinants of the 24 2 X 2 matrices whose entries are all permutations of 1,2,3,4 can only assume the values -10,-5,-2,2,5,10.

Cf. A085000, A088214, A088215, A088216, A088238, A325900.

Fortran
```
C See link given in A088238.
```

f[n_] := (p = Permutations[ Table[i, {i, n^2}]]; Length[ Union[ Table[ Det[ Partition[ p[[i]], n]], {i, 1, (n^2)!}]]]) (* Robert G. Wilson v *)

Minor edits and a(6) from Hugo Pfoertner, Sep 08 2019

A327281 Numbers less than the maximum possible determinant A301371(8)=27296640 not occurring as determinant of an 8 X 8 matrix whose entries are a permutation of the multiset {1^8,..,8^8}.

Original entry on oeis.org

27003797, 27011623, 27012187, 27012757, 27012835

Offset: 1

Hugo Pfoertner, Sep 20 2019

The sequence terms are based on numerical results. No proof for the non-existence of a matrix with given determinant value less than Gasper's upper bound (see Corollary 3 in Sigg) is known. The number of sequence terms is <= 205426. Candidates for a continuation of the sequence are provided as external file.

			The following matrices have determinants in the vicinity of a(1) = A322576(8) = 27003797, for which no corresponding matrix is known:
27003795 = det[2,5,1,4,8,7,3,6; 3,2,4,8,5,5,8,2; 7,1,7,3,4,8,2,4; 8,7,1,4,2,5,6,3; 1,6,7,3,1,6,6,6; 4,8,7,5,6,3,2,1; 5,3,5,1,7,1,7,6; 5,4,4,8,3,2,2,8],
27003796 = det[1,5,6,3,7,4,8,2; 6,4,8,2,2,8,3,3; 4,1,2,3,6,7,5,8; 5,5,2,8,6,7,3,1; 8,6,2,3,2,3,8,4; 1,6,5,7,1,4,4,7; 5,8,4,2,7,3,1,6; 6,1,7,7,5,1,4,5],
27003798 = det[7,4,2,8,7,3,1,5; 3,6,6,1,8,2,4,6; 2,1,3,5,6,8,6,5; 6,5,7,3,3,8,1,3; 5,2,8,6,4,2,7,2; 8,3,3,2,2,4,6,8; 1,7,5,7,1,4,4,7; 5,8,1,4,4,5,7,1],
27003799 = det[2,8,6,4,7,1,5,4; 5,7,1,8,1,4,6,4; 5,3,7,3,3,6,8,1; 2,3,8,6,2,5,2,7; 3,4,2,1,5,7,6,8; 8,7,5,2,4,6,1,4; 3,3,3,7,8,7,2,2; 8,1,4,5,6,1,5,6].

Table of n, a(n) for n = 1..5
Hugo Pfoertner, Plot of sequence.
Hugo Pfoertner, Conjectured continuation of A327281, (2019).
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A301371, A322576, A325900.

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

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A088216 Smallest nonnegative number not expressible as determinant of an n X n matrix with elements 1..n^2.

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A088217 Number of distinct values that can be assumed by the determinant of an n X n matrix whose entries are all permutations of the numbers 1..n^2.

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A327281 Numbers less than the maximum possible determinant A301371(8)=27296640 not occurring as determinant of an 8 X 8 matrix whose entries are a permutation of the multiset {1^8,..,8^8}.

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