Minimizing Transpositions in Permutations with Indistinct Variables

Abstract

In any permutation within a symmetric group, the minimum number of transpositions required to return the permutation to the identity can be algorithmically determined. In fact, this algorithm determines this required number of transpositions based on the degree of the group and the cycle structure of the given permutation. This consistent property changes as the set acted upon by Sn becomes a multiset. In this paper, we explore how introducing these duplicates of some of the variables may reduce the required number of transpositions and determine the altered algorithm for this change in required number of transpositions. We find that this reduction is reliant on the number of indistinct variables for each unique variable, as well as the distribution of variables in the permutation and the cycles within the permutation. Since there are more intricacies to this algorithm than the base case of strictly unique variables, for any given cycle structure there are multiple possibilities of the required number of transpositions to return all variables to their original position.