  The mixed braid groups are the subgroups of Artin braid groups whose elements
preserve a given partition of the base points. We prove that the centralizer of
any braid can be expressed in terms of semidirect and direct products of mixed
braid groups. Then we construct a generating set of the centralizer of any
braid on n strands, which has at most k(k+1)/2 elements if n=2k, and at most
$k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, due to work
of N.V.Ivanov and of S.J.Lee. Finally, we describe how one can explicitly
compute this generating set.
