Q:

Prove $P_{n,n-i} = P_{n+1,n-i+1}$

Let $i,n \in \mathbb{N}$ with $n \geq 2i$.
In our combinatorics script it is written that  for the number of different, random partitions it holds that
$$P_{n,n-i} = P_{n+1,n-i+1}$$
But how can one prove that?
I only know that

but that doesn't help me...

A:

Think about this problem intuitively. So let X be a set consisting on n elements and then we can partition this set into n-i non-empty and non-overlapping subsets. Now created another set Y consisting of one more element that X, so let this set be called Y. Now Y has n+1 elements. We can write Y as the disjoint union of X and the extra element. So the total number of partitions of Y into n-i+1 subsets (where the +1 represents the extra element) is the same as the number of partitions of X into n-i non-empty and non-overlapping subsets. 

