Abstract: Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka–Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category \mathcal{NC} of non-crossing partitions and the category \mathcal{P} of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by \mathcal{NC} together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at N=6, there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from \mathcal{NC} or $\latex mathcal{P}$ at moments of order six.

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