co.combinatorics - A positive formula for the dimensions of homogeneous components of free Lie algebras

This doesn't answer the question, but might still be of interest to you. Let $V$ be the $n$-dimensional vector space spanned by your $n$ letters.

The vector space $V^{otimes k}$ has a natural $S_k$ action. There exists an $S_k$ module, which I will denote $text{Lie}(k)$, such that the $k$th homogenous component of the free Lie algebra on $V$ is isomorphic to

$V^{otimes k} otimes_{S_k} text{Lie}(k)$.

And this module has dimension $(k-1)!$. This wont help you with the dimensions you want, but I think that it's interesting.

If you want to read more then you need to learn about operads, and in particular the Lie operad.

If you just want to know the $S_k$-module structure on $text{Lie}(k)$ then it can be given as follows:
Let $C_k$ be a subgroup of $S_k$ generated by a $k$-cycle. Let $W$ be a 'primitive' representation of $C_k$. (this requires a primitive $k$th root of unity in your field).
Then the module we are looking for is $W$ induced up to $S_k$.

This last bit is a bit mysterious to me.