Noncrossing Partitions Under Rotation and Reflection

We consider noncrossing partitions of [n], taken up to rotation and/or reflection. Taken up to rotation, we exhibit a bijection to bicolored plane trees on n edges, and consider its implications. Taken up to reflection, we show that they are counted by the central binomial coefficients and that, somewhat surprisingly, the same count holds when they are taken under both rotation and reflection. The proof uses a pretty involution originating in work of Germain Kreweras. We conjecture that the “same count” phenomenon still holds when the word “noncrossing” is omitted.