Crossings and Nestings in Set Partitions of Classical Types

We study bijections { Set partitions of type X } −̃→ { Set partitions of type X } for X ∈ {A,B,C,D}, which preserve openers and closers. In types A, B, and C, they interchange • either the number of crossings and of nestings, • or the cardinalities of a maximal crossing and of a maximal nesting. In type D, the results are obtained only in the case of non-crossing and nonnesting set partitions. In all types, we show in particular that non-crossing and a non-nesting set partition are essentially uniquely determined by its openers and closers. Set partitions for classical types (via intersection lattices) •A set partition of type An−1 is a usual set partition of [n]: