  Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a full
transformation $f:X\to X$ preserves (or stabilizes) the partition $\mathcal{P}$
if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that
$Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full
transformations of $X$ that preserve the partition $\mathcal{P}$.
  In 2005 Huisheng found an upper bound for the minimum size of the generating
sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of
its parts have the same size. In addition, Huisheng conjectured that his bound
was exact. In 2009 the first and last authors used representation theory to
completely solve Hisheng's conjecture.
  The goal of this paper is to solve the much more complex problem of finding
the minimum size of the generating sets of $T(X,\mathcal{P})$, when
$\mathcal{P}$ is an arbitrary partition. Again we use representation theory to
find the minimum number of elements needed to generate the wreath product of
finitely many symmetric groups, and then use this result to solve the problem.
  The paper ends with a number of problems for experts in group and semigroup
theories.
