A301371 -id:A301371 - cp's OEIS frontend

A309984 Number of n X n Latin squares with determinant 0, divided by 2.

0, 0, 0, 16, 0, 2088, 5752, 199600889

Offset: 1

Hugo Pfoertner, Aug 26 2019

			a(4)=16: There are 2*a(4) = 32 4 X 4 Latin squares with determinant = 0, one of which is
  [1  4  3  2]
  [4  1  2  3]
  [3  2  1  4]
  [2  3  4  1].
An example of a 6 X 6 Latin square with determinant = 0 is
  [1  3  4  6  5  2]
  [3  2  6  5  4  1]
  [4  6  3  2  1  5]
  [6  5  1  3  2  4]
  [5  4  2  1  3  6]
  [2  1  5  4  6  3].

Brendan McKay, Latin squares.
Hugo Pfoertner, Occurrence counts of determinant values of Latin squares for n=1..8, zipped (2019).

Cf. A040082, A136609, A221976, A301371, A308853, A309258, A309985.

A328031 Upper bound for 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, 1, 3, 18, 172, 2343, 42439, 976050, 27583338, 934173632, 37180409223, 1711870023666, 90007747560742, 5346164992890599, 355442084718552178, 26244000000000000000, 2137205155719002036203, 190811368062993357765186, 18577775646585813239195436, 1963166636163973976912956096

Offset: 0

Hugo Pfoertner, Oct 02 2019

Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A301371, A309985, A328030.

for(n=1,20,print1(floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)),", "))

a(n) = floor(n^n*((n+1)/2)*((n+1)/12)^((n-1)/2)) (Corollary 3 in M. Sigg's article).

A301533 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; denominator.

Original entry on oeis.org

1, 12, 15120, 389188800, 64117105007155200, 10347762119105166852096000, 1389578338099539041754702978576000000, 4713072346356421489071058466945878500353772748800000000

Offset: 1

Hugo Pfoertner, Mar 23 2018

The maximum determinant achievable by arranging the fractions 1/1, 1/2, 1/3, ..., 1/n^2 as matrix entries is provided as fraction A301532(n) / a(n).

			See A301532.

Cf. A085000, A301371, A301532 (corresponding numerators).

A309257 a(n) is the minimum positive value of the determinants of circulant order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 3087, 196, 1440, 405, 6325, 726, 7488, 1183, 11025, 1800

Offset: 1

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

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

			For n=2, a(2)=3 is the minimum absolute value determinant of a back circulant Latin square of order 2. An example of one such matrix is [[2 1], [1 2]].
For n=5, a(5)= 75 is the minimum absolute value determinant of a back circulant Latin square of order 5. An example of one such matrix is [[1, 2, 4, 5, 3], [3, 1, 2, 4, 5], [5, 3, 1, 2, 4], [4, 5, 3, 1, 2], [2, 4, 5, 3, 1]] has determinant 75.

Froylan Maldonado, Minimum absolute value determinant of back circulant Latin squares code

Cf. A301371, A308853, A309088, A328062.

Sage
```
See Maldonado link.
```

Modified title and a(8)-a(13) from Hugo Pfoertner, Oct 01 2019
a(14) from Hugo Pfoertner, Oct 07 2019
a(15) from Hugo Pfoertner, Oct 13 2019

A309089 a(n) is the number of absolute values of determinants that come from a unique isotopy class of order n Latin squares.

Original entry on oeis.org

1, 1, 1, 2, 5, 85, 124

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini_, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 11 2019

We apply every symbol permutation on the representatives of isotopic classes to generate Latin square of order n and calculate the determinants.

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

			For n=4, the determinants 80 and 160 are produced by a unique isotopic class which has [[1, 2, 3, 4], [2, 4, 1, 3], [3, 1, 4, 2], [4, 3, 2, 1]] as a representative. All other determinants are produced by multiple isotopic classes. Therefore a(4)=2.

Peterson Lenard, Number of unique determinants
Brendan McKay, Latin squares

Cf. A301371, A308853, A309088.

Sage
```
# See Maldonado link
```

A309259 a(n) is the greatest common divisor of the determinants of order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 63, 196, 144, 405

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 calculate the determinants. We then compute the greatest common divisor of the values obtained.

These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.

			For n=4, the set of absolute values of the determinants is {0, 80, 160}, so the greatest common divisor of the determinants is 80. Therefore, a(4)=80.

Peterson Lenard, Greatest Common Divisor of all determinants
Brendan McKay, Combinatorial Data

Cf. A301371, A308853, A309088, A309258.

Sage
```
# See Peterson Lenard link
```

a(8), a(9) from Hugo Pfoertner, Sep 02 2019

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

A309984 Number of n X n Latin squares with determinant 0, divided by 2.

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A328031 Upper bound for the determinant of an n X n matrix whose entries are a permutation of the multiset {1^n,...,n^n}.

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A301533 Maximum determinant of an n X n matrix with entries 1, 1/2, .., 1/n^2; denominator.

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A309257 a(n) is the minimum positive value of the determinants of circulant order n Latin squares.

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A309089 a(n) is the number of absolute values of determinants that come from a unique isotopy class of order n Latin squares.

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Sage

A309259 a(n) is the greatest common divisor of the determinants of order n Latin squares.

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Sage

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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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