A088021 -id:A088021 - cp's OEIS frontend

A309258 a(n) is the number of distinct absolute values of determinants of order n Latin squares.

1, 1, 1, 3, 6, 197, 3684, 159561

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 19 2019

We apply every symbol permutation on the representatives of isotopic classes to generate Latin squares of order n and calculated the determinants. We then obtained the absolute values of the determinants and removed duplicates.
These results are based on work supported by the National Science Foundation under grants numbered DMS-1852378 and DMS-1560019.
a(9) >= 1747706. - Hugo Pfoertner, Nov 20 2019

			For n = 5, the set of absolute values of determinants is {75, 825, 1200, 1575, 1875, 2325}, so a(5) = 6.

Froylan Maldonado, Code
Brendan McKay, Latin squares
Hugo Pfoertner, 8X8 Latin squares: Illustration of occurrence counts of determinant values, 5.7 MB, zoom in to see details (2019).
Hugo Pfoertner, Occurrence counts of determinant values for n=1..8, zipped (2019).

Cf. A040082, A088021, A301371, A308853, A309984, A309985.

Sage
```
# See Maldonado link.
```

a(8) from Hugo Pfoertner, Aug 26 2019

A136609 (1/(n!)^2) * number of ways to arrange the consecutive numbers 1...n^2 in an n X n matrix with determinant = 0.

Original entry on oeis.org

0, 0, 76, 14392910

Offset: 1

Hugo Pfoertner, Jan 21 2008

The computation of a(5) seems to be currently (Jan 2008) out of reach (compare with A088021(5)).

			a(1)=0 because det((1))/=0, a(2)=0, because the only possible determinants of a matrix with elements {1,2,3,4} are +-2, +-5 and +-10.

Cf. A001044, A046747, a(3)=A088215(0), a(4)=A136608(0), A221976.

A301372 Conjectured best solution for the problem stated in A098072.

Original entry on oeis.org

0, 1, 20, 54, 61, 87, 89, 97, 99

Offset: 1

Hugo Pfoertner, Mar 26 2018

In A098072 an example of a 3 X 3 matrix was shown that produces the maximum number A088021(3)=10080 of distinct determinants when all permutations of the given matrix entries are performed under the condition of minimizing the greatest entry of the matrix. The current sequence improves this result, i.e., the maximum is reduced from 100 to 99. It is optimal under the assumptions that the matrix entries are distinct and that the two smallest entries are 0 and 1.

The resulting determinant values are given in A301757.

Cf. A088021, A098072, A301757.

A301757 Positive determinant values assumed by performing all permutations of entries in the 3 X 3 matrix of A301372.

Original entry on oeis.org

27, 147, 168, 171, 197, 293, 317, 331, 332, 408, 441, 469, 532, 547, 568, 643, 717, 819, 845, 901, 909, 971, 1017, 1028, 1080, 1104, 1182, 1201, 1297, 1388, 1392, 1400, 1423, 1591, 1606, 1624, 1633, 1640, 1846, 1891, 2038, 2042, 2089, 2114, 2275, 2278, 2288, 2369, 2384

Offset: 1

Hugo Pfoertner, Mar 26 2018

A 3 X 3 matrix with given 9 matrix entries can produce A088021(3)=10080 distinct determinants if all positional permutations are performed. The current sequence provides the 5040 positive determinants of a conjectured optimal matrix minimizing its greatest matrix entry.

			a(1) = 27 because the smallest determinant that can be achieved from the matrix entries of A301372 is
det (( 0  1 89)
     (87 99 97)
     (54 61 20)) = 27,
.
a(5040) = 1039208:
det ((99 54  1)
     (20 97 87)
     (61  0 89)) = 1039208.

Hugo Pfoertner, Table of n, a(n) for n = 1..5040
Hugo Pfoertner, Determinant values plotted as CDF.

Cf. A088021, A301372.

A098072 An example of a 3 X 3 matrix with nonnegative elements that produces the maximum possible number of 10080 different determinants if all 9! permutations of the matrix elements are performed. The target is to find a matrix for which the largest element becomes as small as possible.

Original entry on oeis.org

0, 1, 17, 43, 82, 87, 88, 91, 100

Offset: 1

Hugo Pfoertner, Nov 19 2004

In November 2004 this is the example with the smallest known largest element. It was found in a random search after 3 CPU (1.5 GHz Intel Itanium 2) months. No improvement was found in another 6 months of CPU time.

Hugo Pfoertner, List of 3 X 3 integer matrices that give 10080 different determinants.
Hugo Pfoertner, Search minimal 3 X 3 matrix with 10080 different determinants. FORTRAN program.

Cf. A088021 maximal number of different determinants of an n X n matrix, A099834 different determinants of matrix with nonnegative entries <=n.
Improved solution: A301372.
Optimal solution found by exhaustive search: A316601.

Fortran
```
! See Links section.
```

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.

A372241 a(n) = Product_{j=1..n} j^(ceiling(sqrt(j))).

Original entry on oeis.org

1, 1, 4, 36, 576, 72000, 15552000, 5334336000, 2731180032000, 1991030243328000, 19910302433280000000, 291506737925652480000000, 6044683717626329825280000000, 172642211659125606139822080000000, 6632223203096969285467405025280000000, 335756299656784070076787379404800000000000

Offset: 0

Vaclav Kotesovec, Apr 23 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms)

Cf. A255322, A372240.

Cf. A088021.

Table[Product[j^(Ceiling[Sqrt[j]]), {j, 1, n}], {n, 0, 15}]

a(n^2) = (n^2)!^(n+1) / A255322(n).
log(a(n)) ~ (2*n^(3/2)/3 + n/2 - sqrt(n)/6 + 1/4)*log(n) - 4*n^(3/2)/9 - n/2 + sqrt(n).
a(n^2) / A372240(n^2) = (n^2)! / n!^2 = A088021(n).

A316601 Optimal solution for the problem stated in A098072.

Original entry on oeis.org

0, 3, 19, 65, 75, 83, 88, 93, 94

Offset: 1

Hugo Pfoertner, Jul 13 2018

The nonexistence of solutions with maximum matrix element < 94 was proved by exhaustive search.

Cf. A088021, A098072, A301372, A301757, A316602.

A316602 Positive determinant values assumed by performing all permutations of entries in the 3 X 3 matrix of A316601.

Original entry on oeis.org

1, 9, 29, 67, 162, 267, 309, 430, 452, 520, 570, 712, 716, 825, 841, 844, 941, 943, 980, 1120, 1287, 1289, 1396, 1478, 1516, 1521, 1580, 1592, 1605, 1700, 1753, 1870, 1875, 1914, 1950, 1989, 2157, 2245, 2254, 2265

Offset: 1

Hugo Pfoertner, Jul 17 2018

			a(1) = 1 because the smallest determinant that can be achieved from the matrix entries of A316601 is
det (( 0  3 19)
     (94 88 93)
     (83 75 65)) = 1,
.
a(5040) = 1039208:
det ((88  3 83)
     (75 94 19)
     ( 0 65 93)) = 1044316.

Hugo Pfoertner, Table of n, a(n) for n = 1..5040

Cf. A088021, A301372, A301757, A316601.

cp's OEIS Frontend

A309258 a(n) is the number of distinct absolute values of determinants of order n Latin squares.

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A136609 (1/(n!)^2) * number of ways to arrange the consecutive numbers 1...n^2 in an n X n matrix with determinant = 0.

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A301372 Conjectured best solution for the problem stated in A098072.

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A301757 Positive determinant values assumed by performing all permutations of entries in the 3 X 3 matrix of A301372.

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A098072 An example of a 3 X 3 matrix with nonnegative elements that produces the maximum possible number of 10080 different determinants if all 9! permutations of the matrix elements are performed. The target is to find a matrix for which the largest element becomes as small as possible.

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Fortran

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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A372241 a(n) = Product_{j=1..n} j^(ceiling(sqrt(j))).

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A316601 Optimal solution for the problem stated in A098072.

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A316602 Positive determinant values assumed by performing all permutations of entries in the 3 X 3 matrix of A316601.

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