Vaughan Jones introduced an index for inclusions of certain von Neumann algebra in the 1980's and proved that it is surpisingly rigid. This rigidity is due to a rich combinatorial structure that is inherent to the representation theory of a subfactor with finite index. Subfactor representations reveal interesting unitary tensor categories, or quantum symmetries, whose algebras of intertwiners always contain the Temperley-Lieb algebras and, if an intermediate subfactor is present, the Fuss-Catalan algebras of Jones and myself. The case of two intermediate subfactors is much more involved and not much progress had been made since the late 1990's.

I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a subfactor when two intermediate subfactors occur, and the four algebras form a cocommuting square. These new symmetries turn out to be related to partition algebras and Bell numbers.