Idempotent generation in the endomorphism monoid of a uniform partition

Denote by ${\mathcal{T}}_n$ and ${\mathcal{S}}_n$ the full transformation semigroup and the symmetric group on the set $\{1,\dots ,n\}$, and $\mathcal E_n={1}\cup(\mathcal T_n\setminus \mathcal S_n)$\mathrm{.}Let$\mathcal T(X,\mathcal P)$denotethesetofalltransformationsofthefiniteset$X$ preserving a uniform partition $\mathcal{P}$ of $X$ into $m$ subsets of size $n$, where $m,n\ge 2$. We enumerate the idempotents of $\mathcal{T}(X,\mathcal{P})$, and describe the subsemigroup $S=⟨E⟩$ generated by the idempotents $E=E(\mathcal{T}(X,\mathcal{P}))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of ${\mathcal{E}}_n$, and $S_2$ is a wreath product of ${\mathcal{T}}_n$ with ${\mathcal{T}}_m\setminus {\mathcal{S}}_m$. We calculate the rank and idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ${\mathcal{E}}_n$.