Counting Functions or Asymptotic Densities for Subsets of k-almost Primes 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?
 A: For a complete answer, see this paper:
Integers with a predetermined prime factorization.
There, the author defines two functions,  $\sigma_{\lambda}(x)$ and $\pi_{\lambda}(x)$ where $\lambda$ is some vector in $\mathbb{N}^k$.  These count the number of integers up to $x$ of the form $p_1^{\lambda_1}\cdots p_r^{\lambda_r}$ where $\pi_{\lambda}(x)$ has the added condition that $p_i\neq p_j$ when $i\neq j$.  The result is general asymptotics for both. 
Also see this Math Stack Exchange question and answer.
Examples:  Using your notation above, we have:
$$\pi_{(1,1,2)}(x)\sim \sum_p P(2) \frac{x \log \log x}{\log x}, $$ where $P(s)=\sum_p \frac{1}{p^s}$ is the prime zeta function.
Another example is:
$$\pi_{(1,1,2,5)}(x)\sim C_{(1,1,2,5)} \frac{x \log \log x}{\log x},$$ where $$C_{(1,1,2,5)}=P(2)P(5)-P(10)=\sum_{p}\frac{1}{p^2}\sum_{p}\frac{1}{p^5}-\sum_{p}\frac{1}{p^{10}}.$$
A: I've been asked to submit this partial answer.
Asymptotically, almost all k-almost primes are squarefree. This follows from Landau's asymptotic
$$
\pi_k(x)\sim\tau_k(x)\sim\frac{x}{\log x}\cdot\frac{(\log\log x)^{k-1}}{(k-1)!}
$$
which holds both for numbers with $\omega(n)=k$ and for numbers with $\Omega(n)=k.$
First, consider $k$-almost primes which are divisible by the square of some prime. They are $\ell$-almost primes for some $\ell<k$ and so there are at most $\pi_\ell(x)$ such numbers. Repeating this for each of the other $p(k)-1$ classes of $k$-almost primes one gets their total density at most $\left(p(k)-1\right)\pi_{k-1}.$ Hence the number of squarefree divisors must make up the whole asymptotic density.
Explicitly, in your notation,
$$
\pi p_{1,\ldots1}\sim\frac{x}{\log x}\cdot\frac{(\log\log x)^{k-1}}{(k-1)!}
$$
where there are $k$ 1s, and
$$
\pi p_\lambda\ll\frac{x}{\log x}\cdot(\log\log x)^{k-2}
$$
if there are any numbers larger than 1 in $\lambda.$
