Bijections between noncrossing and nonnesting partitions for classical reflection groups

Alex Fink (1) , Benjamin Iriarte Giraldo (2)

1 Department of Mathematics [Berkeley]
2 UNIANDES - Universidad de los Andes [Bogota]

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.

Mots clés: noncrossing partition; nonnesting partition; reflection group; catalan number; coxeter group

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Alex Fink, Benjamin Iriarte Giraldo. Bijections between noncrossing and nonnesting partitions for classical reflection groups. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.397-410, ⟨10.46298/dmtcs.2737⟩. ⟨hal-01185429⟩