  We investigate the class of quasitrivial semigroups and provide various
characterizations of the subclass of quasitrivial and commutative semigroups as
well as the subclass of quasitrivial and order-preserving semigroups. We also
determine explicitly the sizes of these classes when the semigroups are defined
on finite sets. As a byproduct of these enumerations, we obtain several new
integer sequences.
