2
$begingroup$
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 $k=2$ .
co.combinatorics partitions
share ...
