I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.


My own calculations are quite slow, even for $n = 8$. I suppose my question is how large could one expect to scale this computation, if one desires an explicit decomposition.


I am aware there are estimates of the parameters known for large $n$ (e.g. see <http://www.maths.qmul.ac.uk/~twm/ConjRep.pdf>). 



Decomposing the conjugacy representation of Sym$(n)$ for small $n$
The multiplicity of the irreducible character of $S\_n$ indexed by the partition $\lambda$ of $n$ in the action of $S\_n$ on itself by conjugation is the coefficient of the Schur function $s\_\lambda$ in the Schur function expansion of $1/(1-p\_1)(1-p\_2)(1-p\_3)\cdots$, where $p\_i$ is a power sum symmetric function. (This is an exercise in Chapter 7 of *Enumerative Combinatorics*, vol. 2.) This makes it very easy to compute the decomposition of the conjugacy action using Stembridge's SF package for Maple. For instance, I computed on my laptop the entire decomposition for $n=20$ in a couple of seconds.


**Note added 1/10/14:** Originally I inadvertently wrote $1/(1-p\_1-p\_2-p\_3-\cdots)$ instead of $1/(1-p\_1)(1-p\_2)(1-p\_3)\cdots$. This has been corrected. The "incorrect" symmetric function $1/(1-p\_1-p\_2-p\_3-\cdots)$ is also interesting. It is equal to 
 $$ \frac{\sum\_{n\geq 0}h\_n}{1-\sum\_{n\geq 1}(n-1)h\_n}. $$
See for instance <http://www.math.miami.edu/~wachs/papers/chrom.pdf> and let $t\to 1$.


