Let $E$ be a finite set of $n$ elements and $(x_i)_{i\in I}$ be a finite family of elements of the set of strictly positive integers such that $n=\sum_i x_i$. Let $A$ be the set of partitions $(Y_i)_{i\in I}$ of $E$ such that $|Y_i|=x_i$ for all $i$.
I read somewhere that the symmetric group of the set $E$, $\mathfrak{S}_E$, acts on the set $A$ transitively. Is this true? What is the homomorphism representation of this action?