The class equation and counting in factorizable monoids

For orders and conjugacy in finite group theory, Lagrange's Theorem and the class equation have universal application. Here, the class equation (extended to monoids via standard group action by conjugation) is applied to factorizable submonoids of the symmetric inverse monoid. In particular, if is a monoid induced by a subgroup of the symmetric group , then the center (all elements of that commute with every element of ) is if and only if is transitive. In the case where is both transitive and of order either or (for prime), formulas are provided for the order of as well as the number and sizes of its conjugacy classes.