This question is an extension of this question. There the asymptotic density of k-almost primes was asked.

By subsets I mean the following: Let $\lambda$ be a partition of $k$ and $P_{\lambda}=\{ \prod p_m^{\lambda_m} \; |\; p_m\neq p_k \}$. So $P_{(1,1)}$ would be all semiprimes, despite squares.

What I got are results on $k$-almost primes, being the union of all subsets $P_{\lambda}$. Here are some explicite formulas, like $$ \pi_2(n)=\sum_{i=1}^{\pi(n^{1/2})}\left[\pi\left(\frac{n}{p_i}\right)-i+1\right]. $$ A general asymptotic is given by $$ \begin{eqnarray*} \pi_k(n) &\sim& \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}\\ \end{eqnarray*} $$ For the case of $P_{(1,1)}$ we just subtract the number of squares from $\pi_2(n)$ and get $$ \pi_{P_{(1,1)}}=\pi_2(n)-\pi(n^{1/2}), $$ but I don't see how to extend this.

So again: How do the counting function $\pi_{P_{\lambda}}(n)$ or their asymptotics look like?