  The symmetric inverse monoid, SIM(n), is the set of all partial one-to-one
mappings from the set {1, 2, ... , n} to itself under the operation of
composition. Earlier research on the symmetric inverse monoid delineated the
process for determining whether an element of SIM(n) has a kth root. The
problem of enumerating kth roots of a given element of SIM(n) has since been
posed, which is solved in this work. In order to find the number of kth roots
of an element, all that is needed is to know the cycle and path structure of
the element. Conveniently, the cycle and cycle-free components may be
considered separately in calculating the number of kth roots. Since the
enumeration problem has been completed for the symmetric group, this paper only
focuses on the cycle-free elements of SIM(n). The formulae derived for
cycle-free elements of SIM(n) here utilize integer partitions, similar to their
use in the expressions given for the number of kth roots of permutations.
