This is a shameless plug, but with Martino Lupini and Laura Mancinska, we recently showed that you can define orbits on ordered pairs for a quantum permutation group, and these will form a coherent configuration just like in the classical case: https://arxiv.org/abs/1712.01820

The corresponding coherent algebra is then the centraliser of the magic unitary defining the quantum permutation group. I wonder if the degrees and multiplicities tell you anything about the quantum permutation group like in the classical case?

]]>