Counting the number of connected components of multi-curves through corresponding permutations

In this paper, we introduce a new method for counting the number of connected components of multi-curves. Our method is based on associating multi-curves with permutations, where we can see that the number of connected components of a multi-curve is directly related to the number of cycles in a cycle decomposition of the corresponding permutation. Our ultimate goal is to obtain a formula about multi-curves that gives the number of connected components of the curves, and obtaining a formula for the number of cycles of a permutation will also accomplish this goal. While we have several naive methods for counting the number of cycles of permutations, none of them gives us such a formula. As an approach, we develop a new combinatorial technique to count the number of cycles of permutations by introducing a new notation of permutation.