A088216 -id:A088216 - 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

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

A088215 (1/36)*number of ways to express n as the determinant of a 3 X 3 matrix with elements 1..9.

Original entry on oeis.org

76, 32, 18, 30, 14, 47, 30, 25, 10, 41, 20, 42, 32, 25, 16, 36, 14, 31, 39, 28, 35, 39, 20, 22, 18, 33, 19, 45, 12, 21, 37, 26, 15, 41, 25, 37, 29, 27, 18, 34, 22, 24, 23, 24, 17, 48, 16, 16, 18, 15, 25, 35, 16, 21, 36, 43, 11, 30, 5, 18, 31, 17, 13, 28, 11, 42, 35, 24, 13, 35

Offset: 0

Hugo Pfoertner, Sep 23 2003

0 can be expressed in 36*76=2716 ways as the determinant of a 3 X 3 matrix which has elements 1..9. One such way is e.g. det ((1 2 3)(4 5 6)(7 8 9))=0. All numbers between -323 and +323 can be expressed by such a determinant. The first number not expressible is given by A088216.

Hugo Pfoertner, Table of n, a(n) for n = 0..412 [Gives all terms]
Hugo Pfoertner, Possible determinants of 3 X 3 matrix with elements 1..9.

Cf. A085000, A088214, A088216, A088217.

Cf. A136608 [analogous sequence for 4 X 4 matrices].

A088214 Numbers less than the maximum possible determinant A085000(3)=412 not occurring as determinant of a 3 X 3 matrix with elements 1..9.

Original entry on oeis.org

324, 329, 355, 357, 358, 362, 364, 365, 367, 373, 375, 378, 381, 383, 386, 387, 394, 397, 399, 401, 403, 406, 409, 411

Offset: 1

Hugo Pfoertner, Sep 23 2003

			398 is not in the sequence because it can be expressed as det ((9 3 5)(4 8 2)(1 6 7)).

Hugo Pfoertner, Possible determinants of 3 X 3 matrix with elements 1..9.

Cf. A085000, A088215, A088216, A088217.

A088237 Numbers less than the maximum possible determinant A085000(4)=40800 not occurring as determinant of a 4 X 4 matrix with elements 1..16.

Original entry on oeis.org

38831, 38875, 38959, 38963, 39013, 39057, 39059, 39061, 39063, 39071, 39099, 39109, 39111, 39125, 39137, 39154, 39155, 39178, 39190, 39191, 39205, 39223, 39245, 39247, 39251, 39254, 39267, 39274, 39277, 39279, 39281, 39297, 39310

Offset: 1

Hugo Pfoertner, Sep 25 2003

Hugo Pfoertner, Table of n, a(n) for n = 1..1073 [gives all terms]

Cf. A085000, A088214, A088216.

Full sequence from Hugo Pfoertner, Aug 31 2014

A325900 Numbers less than the maximum possible determinant A085000(6)=1865999570 not occurring as determinant of a 6 X 6 matrix with entries {1,..,36}.

Original entry on oeis.org

1859163031, 1859166733, 1859193211, 1859235497, 1859254067, 1859268659, 1859282869, 1859288597, 1859291519, 1859294309, 1859309245, 1859317037, 1859320819, 1859324083, 1859324501, 1859331797, 1859333683, 1859335879, 1859348273, 1859348639, 1859351059, 1859358869

Offset: 1

Hugo Pfoertner, Sep 07 2019

There are 4521437 terms in the sequence.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A085000, A088214, A088216, A088237, A088238, A322576.

A088238 Numbers less than the maximum possible determinant A085000(5)=6839492 not occurring as determinant of a 5X5 matrix with elements 1..25.

Original entry on oeis.org

6773999, 6774223, 6774529, 6775471, 6775491, 6775877, 6776023, 6776291, 6776373, 6776557, 6776779, 6776803, 6777487, 6777655, 6777718, 6777731, 6778001, 6778103, 6778111, 6778781, 6778909, 6779065, 6779123, 6779261

Offset: 1

Hugo Pfoertner, Oct 03 2003

The first term of this sequence is A088216(5).

The sequence contains exactly 38298 terms.

Hugo Pfoertner, Table of n, a(n) for n = 1..38298 [Gives all terms]
Hugo Pfoertner, Find all possible values of det(A) for matrix with elements 1..n^2. FORTRAN program.

Cf. A085000, A088214, A088237, A088216, A088217.

Fortran
```
c See link.
```

Full sequence from Hugo Pfoertner, Aug 31 2014

A322576 Least nonnegative integer that cannot be expressed as the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n, ..., n^n}.

Original entry on oeis.org

0, 1, 9, 139, 2111, 40021, 942937, 27003797

Offset: 1

Hugo Pfoertner, Aug 29 2019

			a(1) = 0 because det[1] = 1.
a(2) = 1 because det[1,1; 2,2] = 0 and det[2,1; 1,2] = 3 are the only determinant values >= 0 that can be made by permuting the matrix entries {1,1, 2,2}.
a(3) = 9, because it is the first missing value in the list of A309799(3) = 13 determinant values corresponding to {1,1,1, 2,2,2, 3,3,3}: 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 15, 18.

Cf. A088216, A301371, A309258, A309799, A327281.

A136608 (1/576)*number of ways to express n as the determinant of a 4 X 4 matrix with elements 1...16.

Original entry on oeis.org

14392910, 1550244, 2188523, 2029381, 2828486, 1905576, 2901300, 1813327, 3097897, 2169409, 2695559, 1697839, 3767494, 1682771, 2548638, 2503246, 3286048, 1684275, 3093051, 1655317, 3500693, 2374117, 2403536, 1619568

Offset: 0

Hugo Pfoertner, Jan 21 2008

0 can be expressed in a(0)*(4!)^2=8290316160 ways as the determinant of a 4 X 4 matrix which has elements 1...16. One such way is e.g. det ((1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16))=0. All numbers between -38830 and +38830 can be expressed by such a determinant. The first number not expressible is given by A088216(4). The largest expressible number is given by A085000(4)=40800.

			a(40800)=1 because the only 4X4 matrices with elements 1...16 with the determinant 40800 are the 576 combinations of determinant-preserving row and column permutations of ((16 6 4 9)(8 13 11 1)(3 12 5 14)(7 2 15 10)).

Hugo Pfoertner, Table of n, a(n) for n = 0..40800
Hugo Pfoertner, Illustration of occurrence counts. (Zoom into diagram to see details)
Hugo Pfoertner, Illustration of occurrence counts. Upper range.

Cf. A088237 [numbers not expressible by 4X4 determinant], A088215, A088216, A085000, A136609.

A309799 Number of distinct nonnegative values that can be assumed by the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,..,n^n}.

Original entry on oeis.org

1, 2, 13, 147, 2162, 40498, 948618

Offset: 1

Hugo Pfoertner, Aug 29 2019

a(8) >= 27091220. - Hugo Pfoertner, Sep 23 2019

			a(2) = 2: 0 = det[1,1; 2,2], 3 = det[2,1; 1,2] are the two possible nonnegative values of the determinant.
a(3) = 13, because
   0 = det[1,2,3; 1,2,3; 1,2,3],  1 = det[2,2,1; 3,2,1; 3,3,1],
   2 = det[3,2,3; 1,2,3; 1,1,2],  3 = det[3,3,3; 1,2,2; 1,1,2],
   4 = det[1,3,3; 2,2,1; 1,3,2],  5 = det[2,2,1; 1,3,3; 1,2,3],
   6 = det[1,3,2; 1,2,3; 2,1,3],  7 = det[1,3,1; 1,2,3; 2,2,3],
   8 = det[1,1,2; 3,3,2; 1,3,2], 12 = det[2,3,1; 2,1,3; 3,1,2],
  13 = det[3,3,1; 1,3,2; 2,1,2], 15 = det[2,1,3; 3,1,1; 2,3,2],
  18 = det[2,3,1; 1,2,3; 3,1,2]
are the 13 possible nonnegative values of the determinant.

Cf. A088216, A088217, A301371, A309258, A322576, A327281.

cp's OEIS Frontend

A085000 Maximal determinant of an n X n matrix using the integers 1 to 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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A088215 (1/36)*number of ways to express n as the determinant of a 3 X 3 matrix with elements 1..9.

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A088214 Numbers less than the maximum possible determinant A085000(3)=412 not occurring as determinant of a 3 X 3 matrix with elements 1..9.

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A088237 Numbers less than the maximum possible determinant A085000(4)=40800 not occurring as determinant of a 4 X 4 matrix with elements 1..16.

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A325900 Numbers less than the maximum possible determinant A085000(6)=1865999570 not occurring as determinant of a 6 X 6 matrix with entries {1,..,36}.

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A088238 Numbers less than the maximum possible determinant A085000(5)=6839492 not occurring as determinant of a 5X5 matrix with elements 1..25.

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A322576 Least nonnegative integer that cannot be expressed as the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n, ..., n^n}.

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A136608 (1/576)*number of ways to express n as the determinant of a 4 X 4 matrix with elements 1...16.

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A309799 Number of distinct nonnegative values that can be assumed by the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,..,n^n}.

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