Enumerating kth Roots in the Symmetric Inverse Monoid

TL;DR

This paper develops formulas to enumerate kth roots of cycle-free elements in the symmetric inverse monoid SIM(n), extending prior work on symmetric groups by focusing on cycle-free components and using integer partitions.

Contribution

It provides the first explicit enumeration formulas for kth roots of cycle-free elements in SIM(n), expanding the understanding of root structures in inverse monoids.

Findings

Derived formulas for counting kth roots of cycle-free elements

Utilized integer partitions in enumeration process

Extended root enumeration from symmetric groups to SIM(n)

Abstract

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.