Bijections between noncrossing and nonnesting partitions for classical reflection groups

Alex Fink, Benjamin Iriarte Giraldo
2010 Portugaliae Mathematica  
We present 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
more » ... 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. • the noncrossing partitions N C(W ), which in their classical (type A) 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 • the nonnesting partitions N N (W ), introduced by Postnikov [9] for all the finite crystallographic reflection groups simultaneously.
doi:10.4171/pm/1869 fatcat:77vcxik2irdzndl3ny7jerlf5u