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

%I A309344 #39 Aug 14 2025 00:57:03
%S A309344 1,1,1,2,2,4,36,74
%N A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares.
%C A309344 We found all transversals in the main class Latin square representatives of order n.
%C A309344 These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.
%C A309344 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
%C A309344 a(9)>=407, a(10)>=463, a(11)>=6437, a(12)>=23715. - _Eduard I. Vatutin_, added Mar 01 2025, updated Aug 14 2025
%H A309344 Brendan McKay, <a href="https://users.cecs.anu.edu.au/~bdm/data/latin.html">Combinatorial Data</a>.
%H A309344 Eduard I. Vatutin, <a href="https://evatutin.narod.ru/spectra/spectra_ls_transversals_all.png">Graphical representation of the spectra</a>.
%H A309344 Eduard I. Vatutin, Proving lists (<a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n1_1_item.txt">1</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n2_1_item.txt">2</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n3_1_item.txt">3</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n4_2_items.txt">4</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n5_2_items.txt">5</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n6_4_items.txt">6</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n7_36_items.txt">7</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n8_74_items.txt">8</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n9_407_known_items.txt">9</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n10_463_known_items.txt">10</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n11_6437_known_items.txt">11</a>, <a href="http://evatutin.narod.ru/spectra/spectrum_ls_transversals_n12_23715_known_items.txt">12</a>).
%H A309344 E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, <a href="https://evatutin.narod.ru/evatutin_ls_trans_num_in_dls.pdf">On the number of transversals in diagonal Latin squares of even orders</a> (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
%H A309344 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A309344 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.
%o A309344 (MATLAB)
%o A309344 %This extracts entries from each column.  For an example, if
%o A309344 %A=[1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16], and if list = (2, 1, 4),
%o A309344 %this code extracts the second element in the first column, the first
%o A309344 %element in the second column, and the fourth element in the third column.
%o A309344 function [output] = extract(matrix,list)
%o A309344 for i=1:length(list)
%o A309344     output(i) = matrix(list(i),i);
%o A309344 end
%o A309344 end
%o A309344 %Searches matrix to find transversal and outputs the transversal.
%o A309344 function [output] = findtransversal(matrix)
%o A309344 n=length(matrix);
%o A309344 for i=1:n
%o A309344     partialtransversal(i,1)=i;
%o A309344 end
%o A309344 for i=2:n
%o A309344     newpartialtransversal=[];
%o A309344     for j=1:length(partialtransversal)
%o A309344         for k=1:n
%o A309344             if (~ismember(k,partialtransversal(j,:)))&(~ismember(matrix(k,i),extract(matrix,partialtransversal(j,:))))
%o A309344                 newpartialtransversal=[newpartialtransversal;[partialtransversal(j,:),k]];
%o A309344             end
%o A309344         end
%o A309344     end
%o A309344     partialtransversal=newpartialtransversal;
%o A309344 end
%o A309344 output=partialtransversal;
%o A309344 end
%o A309344 %Takes input of n^2 numbers with no spaces between them and converts it
%o A309344 %into an n by n matrix.
%o A309344 function [A] = tomatrix(input)
%o A309344 n=sqrt(floor(log10(input))+2);
%o A309344 for i=1:n^2
%o A309344     temp(i)=mod(floor(input/(10^(i-1))),10);
%o A309344 end
%o A309344 for i=1:n
%o A309344     for j=1:n
%o A309344         A(i,j)=temp(n^2+1-(n*(i-1)+j));
%o A309344     end
%o A309344 end
%o A309344 A=A+ones(n);
%o A309344 end
%Y A309344 Cf. A003090, A090741 (maximum number), A091323 (minimum number), A301371, A308853, A309088, A344105 (version for diagonal Latin squares).
%K A309344 nonn,hard,more
%O A309344 1,4
%A A309344 _Alvaro R. Belmonte_, _Eugene Fiorini_, _Peterson Lenard_, _Froylan Maldonado_, _Sabrina Traver_, _Wing Hong Tony Wong_, Jul 24 2019