A088217 -id:A088217 - 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

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.

A221976 The number of n X n matrices with zero determinant and with entries a permutation of [1,2,..,n^2].

Original entry on oeis.org

0, 0, 2736, 8290316160

Offset: 1

R. J. Mathar, May 12 2013

This counts a subset of all (n^2)! = A088020(n) matrices which contain elements which are a permutation of [n^2]. The range of determinants is characterized in A085000, and the size of the set of different determinants in A088217.

Because any combination of row and column permutation of matrices with distinct elements generates (n!)^2 = A001044(n) different matrices, and because these restricted permutations leave the (absolute value of) the determinant constant, a(n) is a multiple of A001044(n). This factor does not yet take into account that matrix transpositions also maintain the values of determinants (and which never can be achieved by row or column permutation).

a(n) = A136609(n)*A001044(n).

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

A099834 Maximum number of different determinants that can be produced by permuting the elements of a 3 X 3 integer matrix with nonnegative entries <= n.

Original entry on oeis.org

5, 15, 53, 109, 209, 351, 573, 811, 1193, 1509, 1971, 2501, 3183, 3769, 4511, 5025, 5641, 6165, 6600, 6964, 7354, 7696, 7960, 8110, 8404, 8606, 8704, 8846, 8962, 9125, 9210, 9284, 9362, 9420

Offset: 1

Hugo Pfoertner, Oct 29 2004

For large values of n it is always possible to find a matrix that produces A088021(3)=10080 different determinants. Examples are given in the link. Currently (October 2004) the smallest known n for which a(n)=10080 is 100. The elements of the corresponding matrix are given in A098072.

			a(10)=1509: A corresponding set of matrix elements is {10,9,9,8,7,5,2,1,0}.

Hugo Pfoertner, List of 3 X 3 integer matrices that give 10080 different determinants.
Hugo Pfoertner, Elements of 3 X 3 matrices with maximal number of different determinants.

Cf. A033431, A088021, A088217, A089472, A099815.

Cf. A099815 largest determinant that can be produced by the optimal set of matrix elements.

A088745 Infinite array read by antidiagonals: for n>=1 let T_n = upper left n X n matrix. Then T_1 = (1), T_n has elements 1..n^2, contains T_{n-1} as n-1 X n-1 upper left submatrix and abs(det(T_n)) is a maximized.

Original entry on oeis.org

1, 3, 4, 8, 2, 6, 11, 7, 9, 15, 24, 16, 5, 12, 17, 26, 18, 13, 14, 25, 35, 48, 36, 22, 10, 23, 27, 37

Offset: 1

Paul D. Hanna, Oct 14 2003

Matrices with elements 1..n^2 that are to have maximum absolute determinant under the following construction. Start with a(1,1)=1, build successive n X n matrices by fixing previous (n-1) X (n-1) matrix and adding matrix elements (n-1)^2+1..n^2 on lower and right border of matrix. Determinants of upper left n X n matrices are: {1, -10, 205, -6300, 276363, -15615642, ...}.
The definition is incomplete since it does not say what to do if there are several possibilities for the new border. - N. J. A. Sloane, Oct 18 2003
Terms computed by Hugo Pfoertner (see link). If we start with either 2 X 2 matrices [1,3][4,2] or [1,4][3,2], initially there seems to be a unique solution for the subsequent enhanced matrices.

			The 7 X 7 subarray is:
[ 1, 3, 8, 11, 24, 26, 48]
[ 4, 2, 7, 16, 18, 36, 39]
[ 6, 9, 5, 13, 22, 30, 40]
[15, 12, 14, 10, 21, 34, 42]
[17, 25, 23, 20, 19, 31, 47]
[35, 27, 29, 33, 32, 28, 45]
[37, 43, 46, 41, 49, 44, 38]

Hugo Pfoertner, Construction of maximal determinants. FORTRAN program.

Cf. A088746 (determinants), A085000, A088217.

A097400 Number of different values that can be assumed by the determinant of a 3 X 3 matrix whose elements are all permutations of the consecutive integers in the range (n-4,n+4).

Original entry on oeis.org

167, 207, 341, 489, 635, 777, 913, 1055, 1163, 1325, 1389, 1573, 1643, 1819, 1867, 2039, 2073, 2229, 2295, 2463, 2471, 2649, 2625, 2843, 2787, 2995, 2917, 3171, 3099, 3309, 3241, 3397, 3379, 3557, 3467, 3707, 3631, 3807, 3731, 3919, 3869, 3999, 3947

Offset: 0

Hugo Pfoertner, Aug 19 2004

nonn

a(j)=5761 for all j>=167.

			a(0)=167 because the only determinants not achievable by permuting the positions of the matrix elements -4...+4 are +-81,+-83 and +-85, i.e. a(0)=2*A097399(0)-6+1=167 (+1 for det=0).
a(1)=207 because all values -102...+102 and +-104 can be represented as determinant of a matrix whose elements are a permutation of the 9 numbers -3...5.

Cf. A097399 corresponding maximum determinant, a(5)=A088217(3).

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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A088216 Smallest nonnegative number not expressible as determinant of an n X n matrix with elements 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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A221976 The number of n X n matrices with zero determinant and with entries a permutation of [1,2,..,n^2].

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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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A099834 Maximum number of different determinants that can be produced by permuting the elements of a 3 X 3 integer matrix with nonnegative entries <= n.

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A088745 Infinite array read by antidiagonals: for n>=1 let T_n = upper left n X n matrix. Then T_1 = (1), T_n has elements 1..n^2, contains T_{n-1} as n-1 X n-1 upper left submatrix and abs(det(T_n)) is a maximized.

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A097400 Number of different values that can be assumed by the determinant of a 3 X 3 matrix whose elements are all permutations of the consecutive integers in the range (n-4,n+4).

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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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