Integer partitions and permutations

Question

I am given the pair $(n, \lambda)$ where $\lambda$ is a partition of $n$ such that 6 is not a part in $\lambda$. I am told to let $\lambda^*$ represent the partition of $n$ conjugate to $\lambda$. Now we are supposing $(n, \lambda)$ has the following property: there exists a $\theta \in S_n$, the set of permutations of $\{1,2, \dots, n\}$, and $\theta^* \in S_n$ such that both $\theta, \theta^*$ have order 6, $\theta$ has cycle-structure $\lambda$, and $\theta^*$ has cycle-structure $\lambda^*$. I am asked to determine the possible values of n. I am not sure where to start here. Any help would be great.

Answer 1

Hint: the partition must consist of some parts of size $2$ and $3$ (in order for the associated permutation to have order $6$), it could also have some parts of size $1$.

The dual partition is obtained by reflecting in the diagonal, & this tableau will need to only have parts of size $1,2,3$ or $6$ (If it does not have $6$, then it must have at least one $2$ and $3$ (in order for the associated permutation to have order $6$)).

What shapes are possible ?