# Noncrossing partitions under rotation and reflection

- Published in 2005
- Added on

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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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### BibTeX entry

@article{Noncrossingpartitionsunderrotationandreflection, title = {Noncrossing partitions under rotation and reflection}, abstract = {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].}, url = {http://arxiv.org/abs/math/0510447v3 http://arxiv.org/pdf/math/0510447v3}, year = 2005, author = {David Callan and Len Smiley}, comment = {}, urldate = {2018-10-27}, archivePrefix = {arXiv}, eprint = {math/0510447}, primaryClass = {math.CO}, collections = {combinatorics,easily-explained,geometry,things-to-make-and-do} }