In this question I asked whether there exists some formula that computes the multiplicity of the irreps occuring in the branching rules from $S_n$ to $C_n = \langle r : r^n = 1 \rangle$ where we embed $C_n$ into $S_n$ by mapping a generator $r$ to an $n$ cycle $(12 \cdots n)$. The answer turned out to be positive, given beautifully by Stephen.
In this question I basically want to the ask the same thing except replace $C_n$ by the dihedral group $D_n = \langle r,s : r^n = s^2 = 1 \rangle$ (also fine is a sort of Young diagram prescription like before).
I am thinking about an embedding such that $r$ maps to an $n$ cycle $(12\cdots n)$ and $s$ maps to a product of transpositions $s \mapsto (1,n)(2,n-1)(3,n-2)\cdots$ (this is in natural in the sense of symmetries of regular polygons). For example, for $n=5$ I am thinking of an embedding of $D_5$ into $S_5$ such that $r \mapsto (12345)$ and $s \mapsto (15)(24)$.
It's possible that the previous proof given by Stephen can be easily amended but truth be told I don't know enough about complex reflection groups.
