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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 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.

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.