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A self-orthogonal Latin square is a Latin square orthogonal to its transpose and a SOLS L is idempotent if L(i,i)=i. The number of distinct SOLS of order n may be determined by multiplying the number of idempotent SOLS of order n by n!.

A self-orthogonal Latin square (SOLS) is a Latin square orthogonal to its transpose. Two SOLS L and L' are (row,column)-paratopic if two permutations, one applied to the rows and columns of L and one applied to the symbol set of L, transforms L into L'. Enumeration of the (row,column)-paratopism classes of self-orthogonal Latin squares was performed via an (almost) exhaustive computerized tree search. A number of pruning rules was used to eliminate (row,column)-paratopisms and generate one SOLS from each (row,column)-paratopism class (a repository of these class representatives may found at www.vuuren.co.za -> Repositories). As validation of the results two different approaches to the search tree was implemented.

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.