Given $n$, $m$ and $k$, I would like to evaluate cycle index of $S_n \times S_m$ for $c_1 = c_2 = ... = c_{nm} = k$. What is the fastest known algorithm to calculate it? For this particular case, is there any polynomial time algorithm? I know that $Z(S_n) = \frac{1}{n}\sum_{i=1}^na_iZ(S_{n-i})$. Maybe there is a similiar recursive formula for $Z(S_n \times S_m)$?

Edit: For better explanation of the question here I give an example where $n = 2$, $m = 3$, $k = 2$.

$Z(S_2) = \frac{1}{2}(a_1^2 + a_2)$

$Z(S_3) = \frac{1}{6}(b_1^3 + 3b_1b_2 + 2b_3)$

$Z(S_2 \times S_3) = \frac{1}{12} c_1^6 + \frac{1}{3} c_2^3 + \frac{1}{6} c_3^2 + \frac{1}{4} c_1^2 c_2^2 + \frac{1}{6} c_6$

And now substituting $c_1 = c_2 = c_3 = c_4 = c_5 = c_6 = k$ gives $\frac{1}{12} k^6 + \frac{1}{4} k^4 + \frac{1}{3} k^3 + \frac{1}{6} k^2 + \frac{1}{6} k$, so the answer I'm looking for is $13$. Note that I'm interested only in the value, not the polynomial itself.