Abstract
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.
Highlights
Introduction and backgroundThe Coxeter-Catalan combinatorics is an active field of study in the theory of Coxeter groups, having at its core the numerological concurrences according to which several independently motivated sets of objects to do with a Coxeter group W have the cardinality n i=1 (h + di)/di, where h is the Coxeter number ofW and d1, . . . , dn its degrees
We present type preserving bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner
The noncrossing partitions N C(W ), which in their classical avatar are a long-studied combinatorial object harking back at least to Kreweras [6], and in their generalisation to arbitrary Coxeter groups are due to Bessis and Brady and Watt [4, 5]; and
Summary
The Coxeter-Catalan combinatorics is an active field of study in the theory of Coxeter groups, having at its core the numerological concurrences according to which several independently motivated sets of objects to do with a Coxeter group W have the cardinality n i=1. Two other papers presenting combinatorial bijections between noncrossing and nonnesting partitions independent of ours, one by Stump [12] and by Mamede [8], appeared essentially simultaneously to it. Both of these limit themselves to types A and B, whereas we treat type D; our approach is distinct in its type preservation and in providing additional statistics characterising the new bijections
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