How to measure the information contained in a partition of set?

Question

Is there any formal way to measure the information contained in a partition of set? Consider the case when $[n]$ is the full set. Intuitively, I would expect that the two extremal partitions $\{[n]\}$ and $\{\{i\}: i \in [n]\}$ contain the least and most information.

Answer 1

Let $n_i$ be the number of elements of "class" $\# i$ with $\sum n_i=N$.

Let $$p_i=\frac{n_i}{N}.$$

In this way your initial set is probabilized.

Take now the classical measure of "disorder" called "entropy" in information theory:

$$E= -\sum p_i \log_2(p_i)$$