Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence using the (classical) configuration model.
How to compute or precisely estimate the probability to obtain a simple graph (no loop, no multi-edge)?
For instance, if the sequence is $1, 1, 2, 2, 3, 3$ (thus $n=6$), then the probability experimentally seems close to $0.016$.
Classical results (Erdös-Gallai and Havel-Hakimi) tell if a given sequence may lead to a simple graph (graphic sequences). Therefore, they tell (in linear time) if the probability under concern is non-zero.
There are beautiful asymptotic results for the probability under concern, see in particular Janson papers and van der Hofstad book.
There are also beautiful results based on properties of the sequence, in particular for regular graphs (all $d_i$ are equal) and bounded sequences (the maximal $d_i$ is lower than a given bound). See in particular McKay and Wormald works.
These results hold for families of graphs and consider limits when $n$ grows to infinity, and/or families of sequences of which properties can be specified (like maximum degree bounded by a power of $n$).
This is not the situation I face: I do have one specific sequence and want to estimate the probability that the configuration model gives a simple graph for this specific sequence.
Is there a way to compute this probability in linear time and space, with respect to $n$?
If, instead of just one sequence, I have two sequences of integers (of same sum) and sample bipartite multi-graphs with the configuration model, what is the probability to obtain a simple bipartite graph (no multi-edge)?