Given two partitions of a dataset into clusters, we can evaluate their similarity via adjusted mutual information (AMI). However, the formula is rather complicated in the general case.
In the specific case where each partition splits the dataset into exactly two clusters with one much smaller than the other, is there a simple formula or approximate formula for AMI?
Concretely, suppose the dataset has $N$ discrete elements. Partition 1 splits the data into a cluster of size $a$ and $N-a$, with $a \ll N$. Partition 2 splits the data into a cluster of size $b$ and $N-b$, with $b \ll N$. The intersection between the smaller cluster of each partition has size $c \ll N$.
In terms of $[N, a, b, c]$, what is an approximate formula for the AMI between partitions 1 and 2?