1. Enumerating kth Roots in the Symmetric Inverse Monoid Christopher W. York Dr. Valentin V. Andreev, Mentor Department of Mathematics October 1, 2014

2. The Symmetric Inverse Monoid

3. Cycle and Path Notation Every element in SIM(n) can be expressed as the product of disjoint paths and cycles Paths map a number to the one next to it and the last number to nothing and are denoted with brackets. For example, [12357] maps 1 to 2, 2 to 3, 3 to 5, 5 to 7, and 7 to nothing Cycles map the last number to the first number and are denoted with parenthesis. For example, (3452) maps 3 to 4, 4 to 5, 5 to 2, and 2 to 3 Length of a path or cycle is the number of numbers in it. For example, [12357] is of length 5.

4. Raising Elements to a Power k

5. Definition of kth Root

6. Previous Research Annin et al. [2] first determined whether an element in the symmetric group, an algebraic structure similar to SIM(n), has a kth root Recently, Annin [1] determined whether an element in SIM(n) has a kth root both papers posed the question of how many kth roots an element has

7. Interlacing Paths

8. The Root Counting Function

9. A Simple Case

10. A slightly More Complex Case

11. An element with two weakly varying lengths

12. Paths of length 1

13. Some Helpful Formulas

14. Further Research Elements with more than two varying lengths Elements with cycles Elements with weakly varying lengths starting with paths length 1 Creating programs to calculate the number roots Thank you for listening!

15. References [1] Annin, S. et al., On k’th roots in the symmetric inverse monoid. Pi Mu Epsilon 13:6 (2012), 321-331. [2] Annin, S., Jansen, T. and Smith, C., On k’th roots in the symmetric and alternating Groups, Pi Mu Epsilon Journal 12:10 (2009), 581-589.