Let Tn be the full transformation semigroup on Xn={1,2,⋯,n}. Let ⩽, putit 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.