Interesting Esoterica

Noncrossing partitions under rotation and reflection

Article by David Callan and Len Smiley
  • Published in 2005
  • Added on
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}
}