# combinatorics – Is there a way to list all partitions of a set by moving one element at a time?

Given a set $$S = {1, 2, ldots, n}$$, imagine that you start from an initial partition $$P^{(0)} = {P^{(0)}_1, P^{(0)}_2, ldots}$$ of $$S$$, then you a create new one $$P^{(1)}$$ by moving only one element from some subset $$P^{(0)}_i$$ to some subset $$P^{(0)}_j$$, and so on. Is it possible to list all partitions of $$S$$ in this way, such that no partition repeats in the list?

It’s possible when $$n=1,2,3$$. I don’t know for $$n=4$$. Any help is appreciated!