  We compute the partition function of the $q$-states Potts model on a random
planar lattice with $p\leq q$ allowed, equally weighted colours on a connected
boundary. To this end, we employ its matrix model representation in the planar
limit, generalising a result by Voiculescu for the addition of random matrices
to a situation beyond free probability theory. We show that the partition
functions with $p$ and $q-p$ colours on the boundary are related algebraically.
Finally, we investigate the phase diagram of the model when $0\leq q\leq 4$ and
comment on the conformal field theory description of the critical points.
