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A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares. 0
1, 1, 1, 2, 2, 4, 36, 74 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

LINKS

Table of n, a(n) for n=1..8.

Brendan McKay, Combinatorial Data

EXAMPLE

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.

PROG

(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

CROSSREFS

Cf. A301371, A308853, A309088.

Sequence in context: A327011 A300361 A257617 * A257618 A088895 A257619

Adjacent sequences:  A309341 A309342 A309343 * A309345 A309346 A309347

KEYWORD

nonn,hard,more

AUTHOR

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

STATUS

approved

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Last modified February 27 10:15 EST 2020. Contains 332304 sequences. (Running on oeis4.)