This question concerns the mixed Braid groups $B_{n,P}$. Suppose $P$ is a partition of $\{1, \ldots, n\}$. Consider the subgroup $S_{n,P}$ of the symmetric group $S_n$ consisting of the permutations which preserve the parts of $P$. There is an epimorphism $\phi: B_n \to S_n$ from the Braid group on $n$ strands $B_n$ to the symmetric group $S_n$. The mixed Braid group for the partition $P$ is defined as $$B_{n,P} = \phi^{-1} S_{n,P}.$$ That is $B_{n,P}$ consists of those braids which preserve the parts of $P$ as a permutation of the endpoints.
A presentation of $B_{n,P}$ is given in S. Manfredini Some subgroups of Artin's braid group (available here). The presentation is given in Theorem 4 where the generators are taken to be $$\sigma_i, \quad 1\leq i < n, \ i \neq h_i, \qquad A_{h_i, h_j+1}, \quad 1 \leq i < j < m.$$
I suspect that there is a typo here and we should have $$\sigma_i, \quad 1\leq i < n, \ i \neq h_i, \qquad A_{h_i, h_j+1}, \quad 1 \leq i \leq j < m.$$ That is we have to allow the elements $A_{h_i, h_i +1}$ to be in the generating set. Otherwise this presentation does not work for the case when $P$ has 2 parts.
Can someone please shed some light on this?
