cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Peterson Lenard

Peterson Lenard's wiki page.

Peterson Lenard has authored 8 sequences.

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • MATLAB
    %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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A301371, A308853, A309088, A309258.

Programs

  • Sage
    # See Peterson Lenard link

Extensions

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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A301371, A308853, A309088.

Programs

  • 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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A301371, A308853, A309088, A328062.

Programs

  • Sage
    See Maldonado link.

Extensions

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

Views

Author

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

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A301371, A308853.

Programs

  • 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

Views

Author

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jun 28 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Sage
    # 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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

Links

Programs

  • MATLAB
    %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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Sage
    # See Maldonado link.

Extensions

a(8) from Hugo Pfoertner, Aug 26 2019

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