A308853 -id:A308853 - cp's OEIS frontend

A309985 Maximum determinant of an n X n Latin square.

1, 1, 3, 18, 160, 2325, 41895, 961772, 26978400, 929587995

Offset: 0

Hugo Pfoertner, Aug 26 2019

a(n) = A301371(n) for n <= 7. a(8) < A301371(8) = 27296640, a(9) < A301371(9) = 933251220.
a(10) = 36843728625, conjectured. See Stack Exchange link. - Hugo Pfoertner, Sep 29 2019
A328030(n) <= a(n) <= A301371(n). - Hugo Pfoertner, Dec 02 2019
It is unknown, but very likely, that A301371(n) > a(n) also holds for all n > 9 - Hugo Pfoertner, Dec 12 2020

			An example of an 8 X 8 Latin square with maximum determinant is
  [7  1  3  4  8  2  5  6]
  [1  7  4  3  6  5  2  8]
  [3  4  1  7  2  6  8  5]
  [4  3  7  1  5  8  6  2]
  [8  6  2  5  4  7  1  3]
  [2  5  6  8  7  3  4  1]
  [5  2  8  6  1  4  3  7]
  [6  8  5  2  3  1  7  4].
An example of a 9 X 9 Latin square with maximum determinant is
  [9  4  3  8  1  5  2  6  7]
  [3  9  8  5  4  6  1  7  2]
  [4  1  9  3  2  8  7  5  6]
  [1  2  4  9  7  3  6  8  5]
  [8  3  5  6  9  7  4  2  1]
  [2  7  1  4  6  9  5  3  8]
  [5  8  6  7  3  2  9  1  4]
  [7  6  2  1  5  4  8  9  3]
  [6  5  7  2  8  1  3  4  9].
An example of a 10 X 10 Latin square with abs(determinant) = 36843728625 is a circulant matrix with first row [1, 3, 7, 9, 8, 6, 5, 4, 2, 10], but it is not known if this is the best possible. - _Kebbaj Mohamed Reda_, Nov 27 2019 (reworded by _Hugo Pfoertner_)

Brendan McKay, Latin squares.
Mathematics Stack Exchange, Maximum determinant of Latin squares, (2014), (2016).

Cf. A040082, A301371, A308853, A309258, A309984, A328029, A328030.

a(9) from Hugo Pfoertner, Aug 30 2019

a(0)=1 prepended by Alois P. Heinz, Oct 02 2019

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

Original entry on oeis.org

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

A309088 a(n) is the number of isotopy classes of order n Latin squares that produce a unique determinant.

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 25

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 squares 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=5, the only isotopic class that produces determinants 825, 1875, and 2325 is the one with [[1, 2, 3, 4, 5] [2, 3, 5, 1, 4], [3, 5, 4, 2, 1], [4, 1, 2, 5, 3], [5, 4, 1, 3, 2]] as a representative, and the only isotopic class that produces determinants 1200 and 1575 is the one with [[1, 2, 3, 4, 5], [2, 4, 1, 5, 3], [3, 5, 4, 2, 1], [4, 1, 5, 3, 2], [5, 3, 2, 1, 4]] as a representative.
Therefore, a(5)=2 since there are two isotopic classes that produce determinants that are unique to that isotopic class.

Froylan Maldonado, Sage code
Brendan McKay, Latin squares
Brendan McKay, Order 4 isotopic classes
Brendan McKay, Order 5 isotopic classes
Brendan McKay, Order 6 isotopic classes
Brendan McKay, Order 7 isotopic classes
Index entries for sequences related to determinants

Cf. A301371, A308853.

Sage
```
See Maldonado link.
```

A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 36, 74

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 24 2019

We found all transversals in the main class Latin square representatives of order n.
These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.
For all spectra of even orders all known values included in them are divisible by 2. For all spectra of orders n=4k+2 all known values included in the corresponding spectra are divisible by 4. - Eduard I. Vatutin, Mar 01 2025
a(9)>=407, a(10)>=463, a(11)>=6437, a(12)>=23715. - Eduard I. Vatutin, added Mar 01 2025, updated Aug 14 2025

			For n=7, the number of transversals that an order 7 Latin square may have is 3, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 41, 43, 45, 47, 55, 63, or 133. Hence there are 36 distinct numbers of transversals of order 7 Latin squares, so a(7)=36.

Brendan McKay, Combinatorial Data.
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
Index entries for sequences related to Latin squares and rectangles.

Cf. A003090, A090741 (maximum number), A091323 (minimum number), A301371, A308853, A309088, A344105 (version for diagonal Latin squares).

%This extracts entries from each column.  For an example, if
%A=[1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16], and if list = (2, 1, 4),
%this code extracts the second element in the first column, the first
%element in the second column, and the fourth element in the third column.
function [output] = extract(matrix,list)
for i=1:length(list)
    output(i) = matrix(list(i),i);
end
end
%Searches matrix to find transversal and outputs the transversal.
function [output] = findtransversal(matrix)
n=length(matrix);
for i=1:n
    partialtransversal(i,1)=i;
end
for i=2:n
    newpartialtransversal=[];
    for j=1:length(partialtransversal)
        for k=1:n
            if (~ismember(k,partialtransversal(j,:)))&(~ismember(matrix(k,i),extract(matrix,partialtransversal(j,:))))
                newpartialtransversal=[newpartialtransversal;[partialtransversal(j,:),k]];
            end
        end
    end
    partialtransversal=newpartialtransversal;
end
output=partialtransversal;
end
%Takes input of n^2 numbers with no spaces between them and converts it
%into an n by n matrix.
function [A] = tomatrix(input)
n=sqrt(floor(log10(input))+2);
for i=1:n^2
    temp(i)=mod(floor(input/(10^(i-1))),10);
end
for i=1:n
    for j=1:n
        A(i,j)=temp(n^2+1-(n*(i-1)+j));
    end
end
A=A+ones(n);
end

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

Original entry on oeis.org

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.

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

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A309985 Maximum determinant of an n X n Latin square.

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

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A309088 a(n) is the number of isotopy classes of order n Latin squares that produce a unique determinant.

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Sage

A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares.

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

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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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Keywords

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Examples

Links

Crossrefs

Programs

Sage

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

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Author

Keywords

Comments

Examples

Links

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Programs

Sage

Extensions