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 $q$.