cycle type of product of permutations.

Question

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.

Answer 1

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.