Noncrossing Partitions Under Rotation and Reflection
We consider noncrossing partitions of [n] under the action of (i) the
reflection group (of order 2), (ii) the rotation group (cyclic of order n)
and (iii) the rotation/reflection group (dihedral of order 2n). First, we
exhibit a bijection from rotation classes to bicolored plane trees on n
edges, and consider its implications. Then we count noncrossing partitions
of [n] invariant under reflection and show that, somewhat surprisingly,
they are equinumerous with rotation classes invariant under reflection.
The proof uses a pretty involution originating in work of Germain Kreweras.
We conjecture that the "equinumerous" result also holds for arbitrary
partitions of [n].
