From m-clusters to m-noncrossing partitions via exceptional sequences

Understand

Let W be a finite crystallographic reflection group.

• The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W.
• Natural bijections between these two sets are known.
• For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss-Catalan number.