the joint distribution of the degrees of an Erdős–Rényi graph Does the joint distribution of the degrees of an Erdős–Rényi graph have a closed form?
 A: It's not clear whether you're taking the degree distribution to be labelled or not. In case of labelled vertices (i.e. where we don't need to worry about graph homomorphisms), the answer is probably not. Here's why:
Let $D=(d_1, d_2, \dots, d_n)$ be any viable degree distribution on $n$ labelled vertices with $\sum_{i=1}^{i=n}d_i=2m$. Let $\mathcal{H}_D$ be the set of all graphs which have degree distribution $D$.
Now suppose $\mathbf{G}$ is an Erdős–Rényi random graph on $n$ vertices with edge probability $p$. If $H \in \mathcal{H}_D$, then 
$$\mathbb{P}(\mathbf{G}=H)=p^m(1-p)^{{n\choose{2}}-m}$$
since all graphs in $\mathcal{H}_D$ have $m$ edges.
Therefore,
$$\mathbb{P}(\mathbf{G} \in \mathcal{H}_D)= \sum_{H \in \mathcal{H}_D} \mathbb{P}(\mathbf{G}=H)$$
$$\mathbb{P}(\mathbf{G} \in \mathcal{H}_D) = |\mathcal{H}_D|\times p^m(1-p)^{{n\choose{2}}-m}$$
As far as I know, there is no closed form for $|\mathcal{H}_D|$ -- although there might be one if we allow for self-edges (in which case the $n\choose 2$ will be replaced by $n^2$).
