In this paper, we study the dependencies of the number of symmetric and doubly symmetric diagonal Latin squares on the order N. Using fast generator of diagonal Latin squares (augmented by symmetry checker), we determined these dependencies for order at most 8. We also found a number of doubly symmetric diagonal Latin squares of orders 12, 16 and 20.

Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism Θ, the cardinality of the set PLSΘ of partial Latin squares which are invariant under Θ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms...

This talk’s focus is on the computational complexity of completing partial Latin squares. Our first goal in this talk is to show that for many special classes of partial Latin squares, we can construct completions for these objects in polynomial time. From here, we will prove the fairly surprising claim that completing an arbitrary partial Latin square is an NP-complete task; so while it is “ea...

In this paper we consider the problem of enumerating diagonal Latin squares of small order. In particular we discuss possible algorithmic approaches to this problem and show our results in this regard. Surprisingly, our research showed that the best algorithm for enumerating diagonal Latin squares consists of a number of fixed loops, and its effectiveness can be significantly increased by caref...