  Compact matrix quantum groups are strongly determined by their intertwiner
spaces, due to a result by S.L. Woronowicz. In the case of easy quantum groups,
the intertwiner spaces are given by the combinatorics of partitions, see the
inital work of T. Banica and R. Speicher. The philosophy is that all quantum
algebraic properties of these objects should be visible in their combinatorial
data. We show that this is the case for their fusion rules (i.e. for their
representation theory). As a byproduct, we obtain a unified approach to the
fusion rules of the quantum permutation group $S_N^+$, the free orthogonal
quantum group $O_N^+$ as well as the hyperoctahedral quantum group $H_N^+$.
