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.

Showing 1-10 of 10 results.

A287644 Maximum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 32, 133, 384, 2241
Offset: 1

Views

Author

Eduard I. Vatutin, May 29 2017

Keywords

Comments

Same as the maximum number of transversals in a Latin square of order n except n = 3.
a(10) >= 5504 from Parker and Brown.
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= A357514(n) <= a(n) <= A090741(n). - Eduard I. Vatutin, added Sep 20 2020, updated Mar 03 2023
a(11) >= 37851, a(12) >= 198144, a(13) >= 1030367, a(14) >= 3477504, a(15) >= 36362925, a(16) >= 244744192, a(17) >= 1606008513, a(19) >= 87656896891, a(23) >= 452794797220965, a(25) >= 41609568918940625. - Eduard I. Vatutin, Mar 08 2020, updated Mar 10 2022
Also a(n) is the maximum number of transversals in an orthogonal diagonal Latin square of order n for all orders except n=6 where orthogonal diagonal Latin squares don't exist. - Eduard I. Vatutin, Jan 23 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A348212((n-1)/2) <= a(n) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021

References

  • J. W. Brown et al., Completion of the spectrum of orthogonal diagonal Latin squares, Lecture notes in pure and applied mathematics, volume 139 (1992), pp. 43-49.
  • E. T. Parker, Computer investigations of orthogonal Latin squares of order 10, Proc. Sympos. Appl. Math., volume 15 (1963), pp. 73-81.

Crossrefs

Extensions

a(8) added by Eduard I. Vatutin, Oct 29 2017
a(9) added by Eduard I. Vatutin, Sep 20 2020

A287645 Minimum number of transversals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 3, 32, 7, 8, 68
Offset: 1

Views

Author

Eduard I. Vatutin, May 29 2017

Keywords

Comments

From Eduard I. Vatutin, Sep 20 2020: (Start)
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A287644(n) <= A090741(n).
A lower bound for odd n is A091323((n-1)/2) <= a(n). (End)
By definition, the main diagonal and antidiagonal of a diagonal Latin square are transversals, so a(n)>=2 for all n>=4 (the two diagonals are the same in the order 1 square and there are no diagonal Latin squares of orders 2 or 3). - Eduard I. Vatutin, Jun 13 2021
All cyclic diagonal Latin squares are diagonal Latin squares, so a(n) <= A348212((n-1)/2) for all orders n of which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 25 2021
a(10) <= 128, a(11) <= 814, a(12) <= 448, a(13) <= 43093, a(14) <= 25720, a(15) <= 215721, a(16) <= 7465984. - Eduard I. Vatutin, Mar 11 2021, updated Feb 12 2025

Examples

			From _Eduard I. Vatutin_, Apr 24 2021: (Start)
For example, diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals (see A287648)
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . . 1 .   . . . 0   3 . . .   . 2 . .
  . . . 2   . . 3 .   . 0 . .   1 . . .
  . 3 . .   2 . . .   . . . 1   . . 0 .
and 4 not diagonal transversals
  0 . . .   . 1 . .   . . 2 .   . . . 3
  . 2 . .   3 . . .   . . . 0   . . 1 .
  . . 3 .   . . . 2   1 . . .   . 0 . .
  . . . 1   . . 0 .   . 3 . .   2 . . .
total 8 transversals. (End)
		

Crossrefs

Extensions

a(8) added by Eduard I. Vatutin, Oct 29 2017
a(9) added by Eduard I. Vatutin, Sep 20 2020

A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 21, 61, 64
Offset: 1

Views

Author

Eduard I. Vatutin, Jun 26 2021

Keywords

Comments

a(n) <= A307164(n) - A307163(n) + 1.
a(n) <= A287764(n).
a(10) >= 98, a(11) >= 145, a(12) >= 259, a(13) >= 200, a(14) >= 362, a(15) >= 536, a(16) >= 792, a(17) >= 685, a(18) >= 535, a(19) >= 447, a(20) >= 1011, a(21) >= 747, a(22) >= 872, a(23) >= 885, a(24) >= 1610, a(25) >= 1677, a(26) >= 1266, a(27) >= 1337, a(28) >= 2795. - Eduard I. Vatutin, Oct 02 2021, updated Mar 02 2025

Examples

			For n=7 the number of intercalates that a diagonal Latin square of order 7 may have is 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, or 30. Since there are 21 distinct values, a(7)=21.
		

Crossrefs

Extensions

a(9) added by Eduard I. Vatutin, Oct 22 2022

A345370 a(n) is the number of distinct numbers of diagonal transversals that a diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 14, 47, 182
Offset: 1

Views

Author

Eduard I. Vatutin, Jun 16 2021

Keywords

Comments

a(n) <= A287648(n) - A287647(n) + 1.
a(n) <= A287764(n).
Conjecture: a(12) = A287648(12) - A287647(12) + 1. - Natalia Makarova, Oct 26 2021
a(10) >= 736, a(11) >= 1344, a(12) >= 17693, a(13) >= 18241, a(14) >= 294053, a(15) >= 1958394, a(16) >= 13715. - Eduard I. Vatutin, Oct 29 2021, updated Mar 02 2025

Examples

			For n=7 the number of diagonal transversals that a diagonal Latin square of order 7 may have is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, or 27. Since there are 14 distinct values, a(7)=14.
		

Crossrefs

Extensions

a(8) added by Eduard I. Vatutin, Jul 15 2021
a(9) added by Eduard I. Vatutin, Oct 20 2022

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

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.
		

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

A350585 a(n) is the number of distinct numbers of transversals that an orthogonal diagonal Latin square of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 25, 295
Offset: 1

Views

Author

Eduard I. Vatutin, Mar 27 2022

Keywords

Comments

An orthogonal diagonal Latin square is a diagonal Latin square with at least one orthogonal diagonal mate. Since all orthogonal diagonal Latin squares are diagonal Latin squares, a(n) <= A344105(n).
a(10) >= 193, a(11) >= 3588, a(12) >= 10465. - updated by Eduard I. Vatutin, Jan 29 2023

Examples

			For n=8 the number of transversals that an orthogonal diagonal Latin square of order 8 may have is 16, 32, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 88, 96, 112, 128, 132, 144, 160, 168, 192, 224, 256, 320, or 384. Since there are 25 distinct values, a(8)=25.
		

Crossrefs

A357514 Minimum number of transversals in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 8, 15, 0, 23, 16, 132
Offset: 1

Views

Author

Eduard I. Vatutin, Oct 01 2022

Keywords

Comments

Orthogonal diagonal Latin squares is a diagonal Latin squares that have at least one orthogonal diagonal mate.
a(10) <= 652, a(11) <= 2091, a(12) <= 6240. - Eduard I. Vatutin, Oct 01 2022, updated Oct 21 2024
Every diagonal Latin square is a Latin square and every orthogonal diagonal Latin square is a diagonal Latin square, so 0 <= A287645(n) <= a(n) <= A287644(n) <= A090741(n). - Eduard I. Vatutin, Feb 17 2023

Crossrefs

A345761 a(n) is the number of distinct numbers of orthogonal diagonal mates that a diagonal Latin squares of order n can have.

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 3, 31, 99
Offset: 1

Views

Author

Eduard I. Vatutin, Jun 26 2021

Keywords

Comments

a(n) <= A287695(n) + 1.
a(n) <= A287764(n).
a(10) >= 10. It seems that a(10) = 10 due to long computational experiments within the Gerasim@Home volunteer distributed computing project did not reveal the existence of diagonal Latin squares of order 10 with the number of orthogonal diagonal Latin squares different from {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.
a(11) >= 112, a(12) >= 5079. - Eduard I. Vatutin, Nov 02 2021, updated Jan 23 2023

Examples

			For n=7 the number of orthogonal diagonal Latin squares that a diagonal Latin square of order 7 may have is 0, 1, or 3. Since there are 3 distinct values, a(7)=3.
		

Crossrefs

A387187 a(n) is the number of distinct numbers of transversals an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 4, 5, 244, 62
Offset: 1

Views

Author

Eduard I. Vatutin, Aug 21 2025

Keywords

Comments

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.

Examples

			For n=8 the number of transversals that an extended self-orthogonal diagonal Latin square of order 7 may have is 128, 192, 224, 256, or 384. Since there are 3 distinct values, a(8)=5.
		

Crossrefs

A382505 a(n) is the number of distinct numbers of diagonal transversals in Brown's diagonal Latin squares of order 2n.

Original entry on oeis.org

0, 1, 2, 20, 349
Offset: 1

Views

Author

Eduard I. Vatutin, Mar 29 2025

Keywords

Comments

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).
Brown's diagonal Latin squares are special case of plain symmetry diagonal Latin squares that do not exist for odd orders.
a(6)>=1785, a(7)>=60341, a(8)>=4151.

Examples

			For n=4 the number of transversals that a diagonal Latin square of order 8 may have is 0, 8, 12, 16, 18, 20, 24, 26, 28, 32, 36, 40, 44, 48, 52, 56, 64, 88, 96, or 120. Since there are 20 distinct values, a(4)=20.
		

Crossrefs

Showing 1-10 of 10 results.