Ranks of certain semigroups of transformations with idempotent complement whose restrictions belong to a given semigroup

For n ≥ 2, let Pn , In , Tn , and Sn be the partial transformation semigroup, symmetric inverse semigroup, (full) transformation semigroup, and symmetric group on the set Xn = {1, . . . , n}, respectively. In this paper, we find the ranks of certain subsemigroups of Pn , In , and Tn consisting of transformations with idempotent complement whose restrictions to the set Xm belong to the (possible) semigroup Sm, Im, Tm, or Pm for 1 ≤ m ≤ n − 1.