Enumerating \(k\)th Roots in the Symmetric Inverse Monoid

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