cp's OEIS Frontend

A160366 Number of transpose-isomorphism classes of self-orthogonal Latin squares of order n.

%I A160366 #10 Mar 31 2012 20:25:00
%S A160366 1,0,0,5,11,0,1986,52060,34564884,440956473828
%N A160366 Number of transpose-isomorphism classes of self-orthogonal Latin squares of order n.
%C A160366 A self-orthogonal Latin square (SOLS) is a Latin square orthogonal to its transpose. Two SOLS L and L' are (row,column)-paratopic if a permutation p applied to the rows and columns of L and a permutation q applied to the symbol set of L transforms L into L', in which case (p,q) is called an (row,column)-paratopism from L to L'. If p=q, then L and L' are transpose-isomorphic. An (row,column)-autoparatopism is an (row,column)-paratopism that maps L onto itself. The number of transpose-isomorphism classes of SOLS of order n may be determined by the formula sum_{L in I(n)} sum_{a in A(L)}y(a)/|A(L)| where I(n) is a set of (row,column)-paratopism class representatives of SOLS of order n, A(L) is the set of (row,column)-autoparatopism of L for which p and q are both of the same type (x_1,x_2,...,x_n) and y(a)=\prod_{i=1}^n x_i!i^{x_i}. A set of (row,column)-paratopism class representatives may be found at www.vuuren.co.za -> Repositories.
%D A160366 G. P. Graham and C.E. Roberts, 2006. Enumeration and isomorphic classification of self-orthogonal Latin squares, Journal of Combinatorial Mathematics and Combinatorial Computing, 59, pp. 101-118.
%H A160366 A. P. Burger, M. P. Kidd and J. H. van Vuuren, 2010. <a href="http://www.vuuren.co.za/papers/SOLSEnum10.pdf">Enumerasie van self-ortogonale Latynse vierkante van orde 10</a>, LitNet Akademies (Natuurwetenskappe), 7(3), pp 1-22.
%H A160366 A. P. Burger, M. P. Kidd and J. H. van Vuuren, <a href="http://www.vuuren.co.za/papers/SOLSEnum.pdf">Enumeration of isomorphism classes of self-orthogonal Latin squares</a>, Ars Combinatoria, 97, pp. 143-152.
%H A160366 M. P. Kidd, <a href="http://www.vuuren.co.za/main.php">A repository of self-orthogonal Latin squares</a>
%Y A160366 Cf. A160365, A160367, A160368.
%K A160366 hard,more,nonn
%O A160366 1,4
%A A160366 _Martin P Kidd_, May 11 2009
%E A160366 Class names corrected, references updated, and a link updated by _Martin P Kidd_, Aug 14 2010