TITLE: Given a random graph $G_{n,p}$, how to get the expectation of number of components with $k$ vertices and $k$ edges?
QUESTION [5 upvotes]: A random graph $G_{n,p}$ is a graph with $n$ vertices, but every edge exists independently  with probability $p$.
If we know 
$$
np = c, 0 < c < 1
$$
how to get the expectation of number of components with $k$ vertices and $k$ edges when $n$ goes to infinity, where $k$ is constant less than $n$.
(In Erdos' classical paper "On the evolution of random graph", he actually gave a result of the problem, but without details of proof.  So I have trouble understanding how he reached the conclusion.)

REPLY [3 votes]: Let $G_k$ be the number of connected graphs with $k$ edges, on $k$ labeled vertices. The probability that any particular $k$ vertices will form a component of the type that interests you is 
$$G_k p^k (1-p)^{\binom{k}{2} - k + k(n-k)}.$$
So your expectation is
$$G_k p^k (1-p)^{\binom{k}{2} - k + k(n-k)} \binom{n}{k}.$$
If $p=c/n$, then this is 
$$G_k (c/n)^k (1-c/n)^{kn + \mbox{constant}} \frac{n^k}{k!} (\mbox{terms going to $1$}) \sim G_k c^k e^{-ck} / k!$$
There is no closed formula for $G_k$; the sequence is A057500 in Sloane. See Joriki's answer for a reasonably nice formula for $G_k$.
I'm not sure that this is what you wanted to ask.