Suppose a set of elements of finite size.
E.g.:
$X = \left\lbrace a,b,c,d,e,f,g \right\rbrace$
There are several ways to partition $X$.
E.g:
$P_1 = \left\lbrace \left\lbrace a,b \right\rbrace, \left\lbrace c \right\rbrace, \left\lbrace d,e,f,g \right\rbrace \right\rbrace$
$P_2 = \left\lbrace \left\lbrace c,f,g \right\rbrace, \left\lbrace a,e \right\rbrace, \left\lbrace b,d \right\rbrace \right\rbrace$
$P_3 = \left\lbrace \left\lbrace a,b,c \right\rbrace, \left\lbrace d,e,f,g \right\rbrace \right\rbrace$
How can one measure the similarity of those partitionings?
In this example, $P_1$ and $P_3$ are similar.
I think I can measure similarity of individual partitions (e.g. using [Jaccard similarity](https://en.wikipedia.org/wiki/Jaccard_index $\rightarrow$ $J(\left\lbrace a,b \right\rbrace, \left\lbrace a,b,c \right\rbrace)=\frac{2}{3}$), but I have no idea on how to use this information to find which partitionings are similar.
Any idea?
Hint:
Given a pair of partitionings $P_1$ and $P_2$, we can construct a weighted bipartite graph from elements of $P_1$ to elements of $P_2$, that is $G=\left\langle P_1 \cup P_3, P_1 \times P_3 \right\rangle$, where each weight is:
$w_{i,j} = J(p_{1,i},p_{2,j}) = \frac{\left\vert\ p_{1,i}\ \cap\ p_{2,j}\ \right\vert}{\left\vert\ p_{1,i}\ \cup\ p_{2,j}\ \right\vert}$
where $p_{1,i} \in P_1$ and $p_{2,j} \in P_2$
The two most similar partitionings are the solution to the maximum weighted bipartite matching problem.
Sorry if my vocabulary is not correct, my background is not in Mathematics.
Based on your suggestion, given two partitions $P_1$ and $P_2$, you could construct a complete bipartite graph, where $P_1$ and $P_2$ are the parts, and assign to every edge the Jaccard similarity weight.
Taking the average weight (for example) of the edges in a maximum matching (which you can find using the augmented path algorithm or max-flow) will give you a number in $[0,1]$.
Partitions also correspond with Young tableaux and conjugacy classes of permutations which could also be used to describe similarity (e.g. distance in the Cayley graph of $S_n$, with transpositions as generators, between representatives of each conjugacy class).