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!