OFFSET
1,4
COMMENTS
A self-orthogonal Latin square (SOLS) is a Latin square orthogonal to its transpose and a SOLS L is idempotent if L(i,i)=i. 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 an (row,column)-paratopism from L to L'. An (row,column)-autoparatopism is an (row,column)-paratopism that maps L to itself. The number of idempotent SOLS of order n may be found by the formula sum_{L in I(n)}2n!/|A(L)|, where I(n) is a set of (row,column)-paratopism class representatives of SOLS of order n and A(L) is the (row,column)-autoparatopism group of L. A set of (row,column)-paratopism class representatives may be found at www.vuuren.co.za -> Repositories.
REFERENCES
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.
LINKS
A. P. Burger, M. P. Kidd and J. H. van Vuuren, 2010. Enumerasie van self-ortogonale Latynse vierkante van orde 10, LitNet Akademies (Natuurwetenskappe), 7(3), pp 1-22.
A. P. Burger, M. P. Kidd and J. H. van Vuuren, Enumeration of isomorphism classes of self-orthogonal Latin squares, Ars Combinatoria, 97, pp. 143-152.
M. P. Kidd, A repository of self-orthogonal Latin squares
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Martin P Kidd, May 11 2009
EXTENSIONS
Class names corrected, references updated, and a link updated by Martin P Kidd, Aug 14 2010
STATUS
approved