  We prove existence of partitions of an open set $\Omega$ with a given number
of phases, which minimize the sum of the fractional perimeters of all the
phases, with Dirichlet boundary conditions. In two dimensions we show that, if
the fractional parameter $s$ is sufficiently close to $1$, the only singular
minimal cone, that is, the only minimal partition invariant by dilations and
with a singular point, is given by three half-lines meeting at $120$ degrees.
In the case of a weighted sum of fractional perimeters, we show that there
exists a unique minimal cone with three phases.
