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

1, 0, 0, 5, 11, 0, 1986, 52060, 34564884, 440956473828

Offset: 1

Author

Martin P Kidd, May 11 2009

Keywords

• hard
• more
• nonn

Comments

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.

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

Cf. A160365, A160367, A160368.

Extensions

Class names corrected, references updated, and a link updated by Martin P Kidd, Aug 14 2010