Alvaro R. Belmonte - cp's OEIS frontend

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

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

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

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
```

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

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.
```

A308853 a(n) is the minimum absolute value of nonzero determinants of order n Latin squares.

Original entry on oeis.org

1, 3, 18, 80, 75, 126, 196, 144, 405

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jun 28 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=2, the only Latin squares of order 2 are [[1, 2], [2, 1]] and [[2, 1], [1, 2]].  Therefore, the minimum absolute value of the determinants of order 2 Latin squares is 3.

Brendan McKay, Latin squares
Hugo Pfoertner, Occurrence counts of determinant values for n=1..8, zipped (2019).
Eric Weisstein's World of Mathematics, Latin square
Wikipedia, Latin square
Index entries for sequences related to determinants

Cf. A040082, A301371 (upper bound for maximum determinant of Latin squares of order n), A309258, A309984, A309985.

# Takes a string and turns it into a square matrix of order n
def make_matrix(string,n):
    m = []
    row = []
    for i in range(0,n * n):
        if string[i] == '\n':
            continue
        if string[i] == ' ':
            continue
        row.append(Integer(string[i]) + 1)
        if len(row) == n:
            m.append(row)
            row = []
    return matrix(m)
# Reads a file and returns a list of the matrices in the file
def fetch_matrices(file_name,n):
    matrices = []
    with open(file_name) as f:
        L = f.readlines()
    for i in L:
        matrices.append(make_matrix(i,n))
    return matrices
# Takes a matrix and permutates each symbol in the matrix
# with the given permutation
def permute_matrix(matrix, permutation,n):
    copy = deepcopy(matrix)
    for i in range(0, n):
        for j in range(0 , n):
            copy[i,j] = permutation[copy[i][j] - 1]
    return copy
"""
Creates a determinant list with the following triples,
[Isotopy Class Representative, Permutation, Determinant]
The Isotopy class representatives come from a file that
contains all Isotopy classes.
"""
def create_determinant_list(file_name,n):
    the_list = []
    permu = (Permutations(n)).list()
    matrices = fetch_matrices(file_name,n)
    for i in range(0,len(matrices)):
        for j in permu:
            copy = permute_matrix(matrices[i],j,n)
            the_list.append([i,j,copy.determinant()])
            print(len(the_list))
    return the_list
# Froylan Maldonado, Jun 28 2019

a(8) from Hugo Pfoertner, Aug 24 2019

a(9) from Hugo Pfoertner, Aug 27 2019

A309345 a(n) is the minimum number of transversals in Latin squares of order n that have at least 1 transversal.

Original entry on oeis.org

3, 8, 3, 8, 3, 8

Offset: 3

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

We find all the transversals of the main class representatives of order n Latin squares then find the one with the fewest transversals.

			For n = 5,they are 2 main classes of Latin squares. One of them has representative M = [[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]], and it has 3 transversals; {M(1,1), M(5,2), M(3,3), M(2,4), M(4,5)}, {M(2,1), M(5,2), M(4,3), M(1,4), M(3,5)}, and {M(4,1), M(5,2), M(2,3), M(3,4), M(1,5)}. The other main class representative [[1,2,3,4,5],[2,3,4,5,1],[3,4,5,1,2],[4,5,1,2,3],[5,1,2,3,4]] has 15 transversals. Therefore, the minimum number of transversals in order 5 Latin squares is 3, i.e., a(5) = 3.

Brendan McKay, 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

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

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User: Alvaro R. Belmonte

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

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A309259 a(n) is the greatest common divisor of the determinants of 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

A309257 a(n) is the minimum positive value of the determinants of circulant 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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Comments

Examples

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Crossrefs

Programs

Sage

A308853 a(n) is the minimum absolute value of nonzero determinants of order n Latin squares.

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Programs

Sage

Extensions

A309345 a(n) is the minimum number of transversals in Latin squares of order n that have at least 1 transversal.

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Author

Keywords

Comments

Examples

Links

Programs

MATLAB

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

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Author

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