I am interested in looking at $n\times n$ tableaux (or matrices) in which (WLOG) each integer in $\{ 1, 2, \ldots, n \}$ occurs exactly $n$ times. This is a generalisation of a Latin (or even semi-Latin) square, which obviously has this property. It is a proper generalisation, as there are tableaux with this property that are not semi-Latin, such as $\left[\begin{matrix} 1&2&2\\3&1&3\\3&1&2\end{matrix}\right].$ I have counted the number of such tableaux, up to a suitable (for my applications) notion of isomorphism for $n = 2,3,4$, and there are lots of them. There are $5$ with $n = 2$; $305$ with $n = 3$; and $2630904$, with $n = 4$. As these generalise Latin squares, it seems like the sort of thing that people in the combinatorics community might have investigated. Have such tableaux been studied before? Do they have a name? (If these have a name, I might do better with Google.) Many thanks.
EDIT: For the $2\times 2$ case, there are $6 = {4\choose 2}$ squares, since we can choose any two of the four matrix positions in which to place a $1$, and then the other two spots must contain a $2$. These are: $\left\{ \left[\begin{matrix}1&1\\2&2\end{matrix}\right], \left[\begin{matrix}1&2\\1&2\end{matrix}\right], \left[\begin{matrix}1&2\\2&1\end{matrix}\right], \left[\begin{matrix}2&1\\1&2\end{matrix}\right], \left[\begin{matrix}2&1\\2&1\end{matrix}\right], \left[\begin{matrix}2&2\\1&1\end{matrix}\right]\right\}.$ Now, of these, only the pair $\left[\begin{matrix}1&2\\2&1\end{matrix}\right]$ and $\left[\begin{matrix}2&1\\1&2\end{matrix}\right]$ are equivalent.
The notion of equivalence or isomorphism used is this. Two such $n\times n$ matrices $(a_{i,j})$ and $(b_{i,j})$ as above, are regarded as essentially the same if there is a permutation $\sigma\in S_{n}$ for which $\sigma( a_{i,j} ) = b_{\sigma i, \sigma j}$, for all $i$ and $j$.