Semigroups of transformations commuting with injective nilpotents

For an injective nilpotent eta in the semigroup PTn of partial transformations on a set with n elements, let C(eta) be the semigroup of all elements of PTn commuting with eta. The structure of C(eta) in terms of Green's relations is determined (including the partial orders of L-, R-, and J-classes), the regular elements of C(eta) are characterized, and a formula for the order of C(eta) is given.