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Equal sized partitions without overlap
There are 24 students in the class and I will divide them into 8 groups with 3 people in each group. In how many different ways can I divide them into these 8 groups so that nobody is ever in a group with someone that they were in a group with before.
More concretely, in the first week I put them in groups $\lbrace 1,2,3 \rbrace, \lbrace 4,5,6 \rbrace, \dots.$ In week two I put them into groups $\lbrace 1,4,7 \rbrace, \lbrace2,5,8 \rbrace, \dots.$ How many weeks can I continue changing the groups and ensure that everyone continues to have new people in his/her group.
I suppose that the general question is let $n$ and $m$ be integers such that $m$ divides $n$ and let $N = \lbrace 1, 2, 3, \dots, n\rbrace.$ Let $P$ denote the set of all partitions of $N$ into $m$ subsets of order $\frac{n}{m}$. Let $Q$ denote a maximal subset of $P$ such that for any partitions $A,B \in Q$ and subsets $A_1 \in A$, and $B_1 \in B$ then $|A_1 \cap B_1|$ equals 1 or 0. (That is, two groups chosen in different weeks have one person in common or nobody in common).