Noncrossing partitions under rotation and reflection

2005/10/20 by David Callan, Callan, David, Len Smiley +1 · 1 citation
Mathematics · #05A15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.math/0510447

openalex publication_date 2005/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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].

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