Alex Fink ; Benjamin Iriarte Giraldo - Bijections between noncrossing and nonnesting partitions for classical reflection groups

dmtcs:2737 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) - https://doi.org/10.46298/dmtcs.2737
Bijections between noncrossing and nonnesting partitions for classical reflection groupsArticle

Authors: Alex Fink 1; Benjamin Iriarte Giraldo 2

We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.


Volume: DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
Section: Proceedings
Published on: January 1, 2009
Imported on: January 31, 2017
Keywords: noncrossing partition,nonnesting partition,reflection group,catalan number,coxeter group,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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