Definitions: a partial Latin rectangle is an $r \times s$ matrix containing symbols from $[n] \cup \{\cdot\}$ such that each row and each column contains at most one copy of any symbol in $[n]$. The cells containing the symbol $\cdot$ are considered empty. The weight $m$ is the number of non-empty cells.
- Partial Latin rectangles will be assumed to have (a) at least one entry in each row, (b) at least one entry in each column, and (c) at least one copy of each symbol.
Autotopism groups and autoparatopism groups are defined analogously to Latin squares (but note that the number of rows, columns, and symbols may not be preserved under paratopisms).
For example, the partial Latin rectangle $$ \begin{bmatrix} 1 & \cdot & 2 \\ \cdot & 1 & \cdot \\ \end{bmatrix} $$ has a trivial autotopism group and an autoparatopism group isomorphic to $C_2$ (swap the last two columns, and apply the row-symbol conjugate). It also has minimum weight among all partial Latin rectangles with those autotopism and autoparatopism groups.
Question: What is the minimum weight of a partial Latin rectangle with autotopism group $\cong C_2$ and autoparatopism group $\cong S_3$?
At this point, I don't know for certain that one even exists, but I don't see any reason it wouldn't. But if it does, it has weight $\geq 11$.
The motivation is that I'm doing a computer search for minimum-weight partial Latin rectangles that have specified autotopism and autoparatopism groups (up to isomorphism). This one didn't come up for weights $m \leq 10$. In a sense, I'm "researching out loud" in the hopes that someone might come up with a clever way of generating these partial Latin rectangles.
