On the idempotents of semigroup of partial contractions of a finite chain

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain. Let $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ be Semigroups of partial and full transformations on $[n]$ respectively. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in \dom~\alpha\}$ and $\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: |x\alpha-y\alpha|\leq|x-y| \ \ \forall x, y\in [n]\}$, then $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ are subsemigroups of $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ respectively. In this paper, we characterize the idempotent elements and computed the number of idempotents of height, $n-1$ and $n-2$ for the semigroups $\mathcal{CP}_{n}$ and $\mathcal{CT}_{n}$ respectively.