I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind enough to do it for me...


Here's the question: partition $n$ into two pieces, $n= p+q$, and let $S\_p\times S\_q \subset S\_n$ be the associated Young subgroup. Now consider the braid group $B\_n$. I'm interested in the subgroup of $B\_n$ consisting of braids that preserve this partition of their endpoints, i.e.
$$ B\_{p,q} := B\_n \times\_{S\_n} (S\_p\times S\_q )$$
I can write down generators for $B\_{p,q}$, namely $s\_1,.., s\_{p-1}, s\_p^2, s\_{p+1}, ..., s\_{n-1}$ where the $s\_i$ are the standard generators of $B\_n$. My question is what are the relations? 


Obviously I need the usual braid relations on each piece of the partition, but are there any others? I'd be happy to extrapolate heuristically from low values of $p$ and $q$. 



Relations in a particular subgroup of the braid group.
You probably already noticed that, but $B\_{p,q}$ is the fundamental group of
$$
X\_n/(S\_p \times S\_q)
$$
where $X\_n$ is the configuration space of $n$ points in the complex plane. Ths may help to guess some facts about these groups.


So far I know these group are usually called "mixed braid groups" in the litterature, though this name is sometimes used for the group of braids with the $p$ first strands fixed. Anyway, a presentation of it is obtained in S. Manfredini **Some subgroups of Artin's braid group** (available [here](http://www.sciencedirect.com/science/article/pii/S0166864196001526)). Results on their relation with the representation theory of $B\_n$, as well as references you may find interesting, can be found in Bellingeri, Godelle, Guaschi, **Exact sequences, lower central series and representations of surface braid groups** ([arXiv](http://arxiv.org/abs/1106.4982)).


