Pairs of Permutations up to Simultaneous Conjugation The conjugacy classes of $S_n$ are the cycle types since if $\tau = (\dots)(\dots)\dots(\dots)$, the conjugation $\tau \mapsto \sigma \tau \sigma^{-1}$ permutes the labels in the cycles of $\tau$.
Has anyone studied pairs of permutations up to simultaneous conjugation $(\tau_1,\tau_2) \mapsto (\sigma \tau_1 \sigma^{-1}, \sigma \tau_2 \sigma^{-1})$?  
These are related to branched covers of a once-punctured torus since $\pi_1(\mathbb{T}-\{ pt\}) = \langle a,b| \text{ no relations }\rangle = \mathbb{F}_2$ we need two generators, $\tau_1, \tau_2$.
 A: Another way to say what's in oeis.org/A110143 is the following. Let
$\lambda$ be a partition of $n$, denoted $\lambda\vdash n$, and let
$z_\lambda$ be the number of permutations in $S_n$ commuting with a
fixed permutation $w$ of cycle type $\lambda$, so
$z_\lambda=1^{m_1}m_1!2^{m_2}m_2!\cdots$, where $\lambda$ has $m_i$
parts equal to $i$. Thus the size of the conjugacy class containing
$w$ is $n!/z_\lambda$.  It is then immediate from the Cauchy-Frobenius
lemma (aka Burnside's lemma) that the number of classes is 
  $$ \frac{1}{n!}\sum_{\lambda\vdash n} \frac{n!}{z_\lambda}z_\lambda^2 = 
     \sum_{\lambda\vdash n} z_\lambda. $$
Similarly the number of $k$-tuples of permutations up to simultaneous 
conjugation is $\sum_{\lambda\vdash n}z_\lambda^{k-1}$. 
A: As you implicitly point out, you might as well ask about conjugacy classes of subgroups of finite index in $F_2$.
You might be interested in a famous paper of Marshall Hall Jr,  in which he computed the number of subgroups of a free group of a given index.  It would be a nice exercise to adapt his argument to compute the number of conjugacy classes. (Or perhaps it's done in the paper?  I don't recall, off the top of my head.)
