block of this partition as $\ker\alpha$-class. In particular, the sets $x\alpha^{-1}=\{y\in X:y\alpha =x\}$, for $x\in X\alpha$, are the $\ker\alpha$-classes. We say that a partition $P$ is a subpartition of a partition $Q$ of $X$ if for all $p\in P$ there is $q\in Q$ with $p\subseteq q$. A set $T\subseteq X$ with $\left\vert T\cap x\alpha^{-1}\right\vert =1$, for all $x\in X\alpha$, is called a transversal of $\ker\alpha$. Let $A\subseteq X$. Then $\alpha |_{A}:A\rightarrow Y$ denotes the restriction of $\alpha$ to $A$ and $A$ will be called convex if $x<y<z$ with $x,z\in A$ implies $y\in A$.

Let $l\in \{1,\ldots ,m\}$. We denote by $\mathcal{P}_{l}$ the set of all partitions $\{A_{1},\ldots,A_{l}\}$ of $X$ such that $A_{2}<A_{3}<\cdots
<A_{l}$ are convex sets (if $l>1$) and $A_{1}$ is the union of two convex sets with $1,n\in A_{1}.$ Further, let $\mathcal{Q}_{l}$ be the set of all partitions $\{A_{1},\ldots,A_{l}\}$ of $X$ such that $A_{1}<A_{2}<\cdots <A_{l}$ are convex and let $\mathcal{R}_{l}$ be the set of all partitions of $X$, which not belong to $\mathcal{Q}_{l} \cup \mathcal{P}_{l}$. We observe that $\ker \beta \in \mathcal{Q}_{l} \cup \mathcal{P}_{l}$, whenever $\beta\in \OP(X,Y)$ with $|X\beta|=l$. In particular, $\ker \beta \in \mathcal{Q}_{l}$, whenever $\beta \in \O(X,Y)$.

Let us consider the case $l=m>1$. For $P\in \mathcal{P}_{m}$ with 