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.