Let $\mathscr{T}_{n}$ be the full transformation semigroup on $X_{n}=\{1, 2, \cdots, n\}$. Let $\leqslant r \leqslant n$, put$\mathscr{F}_{(n, r)}=\left\{\alpha \in \mathscr{T}_{n}: i \alpha=i, \quad \forall i \in\{1, 2, \cdots, r\}\right\}, $it is obvious that the semigroup $\mathscr{F}_{(n, r)}$ is subsemigroup of $\mathscr{T}_{n}$. In the paper, we study the core $\left(\mathscr{C} \mathscr{F}_{(n, r)}\right)=\left\langle E\left(\mathscr{F}_{(n, r)}\right)\right\rangle$ of the semigroup $\mathscr{F}_{(n, r)}$, where $\mathscr{C}\left(\mathscr{F}_{(n, r)}\right)=\left\{\alpha \in \mathscr{F}_{(n, r)}: \alpha^{2}=\alpha\right\}$, by analyzing idempotents of the semigroup $\mathscr{F}_{(n, r)}$, we prove that the rank and idempotent rank of semigroup $\mathscr{C} \mathscr{F}_{(n, r)}$ are both equal to $\frac{(n-r)(n-r-1)}{2}+r(n-r)+1$.
