I am looking for a formula for determining the expected number of independent sets of size $k$ (for arbitrary $k$) in a random graph $G(n,p)$. Here $n$ is the number of vertices and each edge is included with independent probability $p$. I would like to be able to calculate this for arbitrary $p$ if possible.
I have come across the article  which provides a formula for the special case $p = 0.5$.
I have also come across the article  which in p. 12 provides a value for some cases other than $p = 0.5$, so I would assume it is known for some $p$ values other than $0.5$.
My questions are:
- Do you know how one shows, or could you provide a reference for the formula shown in  for the case $p = 0.5$. The paper gives some references but they are about clique problems and I am not sure how I could arrive from them to the result shown in that paper.
- Is there a known formula for arbitrary $p$? If not, for what $p$ values is a formula known and where could I find such formulas?