There may be a theory that deals with problems like this but I'm not
enough of a mathematician to know what it is. So far I've looked up
braid groups, block design, and the recommended related posts to this
one but the penny hasn't dropped yet.

Let's say I'm trying to organize a speed dating event for androgynous
aliens whose "couples" can have arbitrarily many members. Suppose I
call a set $K$ of partitions on a finite set $A$ of aliens a "klatsch"
over $A$ if and only if for any distinct $a,b\in A$ there is a
partition $P\in K$ containing exactly one class $C\in P$ containing
both $a$ and $b$, and no such class in any other partition in $K$. For
example, the set of three partitions
$K=\{\{\{a,b\},\{c,d\}\},\{\{a,d\},\{b,c\}\},\{\{a,c\},\{b,d\}\}\}$ is
a klatsch over $\{a,b,c,d\}$.

If two klatsches over $A$ are considered isomorphic to each other whenever
a bijection from $A$ to $A$ transforms one to the other, then
$\{\{\{a\},\{b,c\}\},\{\{b\},\{a,c\}\},\{\{c\},\{a,b\}\}\}$ is the
only klatsch over a three element set up to isomorphism. My best efforts
at an exhaustive search indicate 5 klatsches over a 4 element set up to
isomorphism (of which one involves a triple), 18 over a 5 element set,
130 over a 6 element set, and so on. Is there an efficient algorithm
for generating all klatsches over a given set up to isomporphism?