Let $C\_1,\dots, C\_m$ be a family of ordered set partitions of $[n]$ with exactly $k$ blocks.


Write $C\_i = \{B\_{i1}, \dots, B\_{ik}\}$ for $i=1,\dots, m$ where $B\_{ij}$ are the blocks of the ordered set partition $C\_i$.


Suppose this family also has the property that for each $j=1,\dots, k$ 


$$B\_{1j} \cup \cdots \cup B\_{mj}$$ 


is also a partition of $[n]$


Can one determine the maximal number of members in such a family $m$, or at least a decent upper bound on $m$?


Edit:


It might also be worth noting that if we take $k=n$, then $m=n$ since this would be equivalent to the existence of a latin square. I am in particular interested in the case $k=2$.



Families of ordered set partitions with disjoint blocks
We have $$mn=\sum\_i\sum\_j |B\_{ij}|=\sum\_j\sum\_i |B\_{ij}|=kn,$$
thus $m=k$. 


