Noncrossing partitions under rotation and reflection

Details Noncrossing partitions under rotation and reflection Noncrossing partitions under rotation and reflection

2005-10-20

arxiv.org/abs/math/0510447v3

Mathematics

Combinatorics

15 pages, LaTeX, PSTricks, .eps figures

Scientific paper

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].

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