The rank of the semigroup of transformations stabilising a partition of a finite set

Let (Formula presented.) be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition (Formula presented.) if for all P ∈ (Formula presented.) there exists Q ∈ (Formula presented.) such that Pf ⊆ Q. Let T(X, (Formula presented.)) denote the semigroup of all full transformations of X that preserve the partition (Formula presented.). In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, (Formula presented.)), when (Formula presented.) is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture. The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, (Formula presented.)), when (Formula presented.) 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.