<s>[INST] Latin squares with one cycle type? Cross posting from MSE, where this question received no answers.
The following Latin square
$$\begin{bmatrix}
1&2&3&4&5&6&7&8\\
2&1&4&5&6&7&8&3\\
3&4&1&6&2&8&5&7\\
4&3&2&8&7&1&6&5\\
5&6&7&1&8&4&3&2\\
6&5&8&7&3&2&4&1\\
7&8&5&2&4&3&1&6\\
8&7&6&3&1&5&2&4
\end{bmatrix}$$
has the property that for all pairs of two different rows $a$ and $b$, the permutations $ab^{-1}$ have the same cycle type (one 2-cycle and one 6-cycle).
What is known about Latin squares with the property that all $ab^{-1}$ have the same cycle type (where $a$ and $b$ are different rows)? For example, do they have a particular structure, for which cycle types do they exist, are there any infinite families known, do they have a name, etc?
The only example of an infinite family I'm aware of are powers of a single cyclic permutation when $n$ is prime, for example:
$$\begin{bmatrix}
1&2&3&4&5\\
2&3&4&5&1\\
3&4&5&1&2\\
4&5&1&2&3\\
5&1&2&3&4
\end{bmatrix}$$
 [/INST] There are also "pan-Hamiltonian" Latin squares, see Perfect Factorisations of Bipartite Graphs and Latin Squares Without Proper Subrectangles by I. M. Wanless, Electronic J. Combin. 6 (1999), R9.
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