# cycle type of product of permutations.

Let $$\sigma$$ and $$\tau$$ be two permutations in $$S_n$$ with partitions $$\lambda$$ and $$\mu$$ as their cycle type.

What is the cycle type of the product $$\sigma \tau$$ in terms of $$\lambda$$ and $$\mu$$?

Thank you.

• I don't think you can tell. For example, $(12)(12) = 1$, but $(12)(23) = (123)$. Both examples are products of two transpositions, but the results have different cycle types. – Nick Aug 3 at 14:43
• hmm. ok. Thanks. Some partial results with some assumptions on $\sigma$ and $\tau$? – GA316 Aug 3 at 15:24

The only intersting case is the one where the partitions are disjoint , i.e. $$c$$ (not a singleton) $$\in \lambda \Rightarrow c \notin \mu$$ and vice versa. The partition of the product is then the union of the partitions, example: $$\sigma = (1,2)(5,6,7)$$ and $$\tau = (3,4)(8,9,10)$$ then $$\sigma\tau=\tau\sigma = (1,2)(3,4)(5,6,7)(8,9,10)$$ in other cases the partitions all get mixed up.