Clusters, diminishing elements and noncrossing partitions

Two well-known classes of objects counted by the Catalan numbers are triangulations of a polygon and noncrossing partitions. More generally, for any finite Coxeter group, there is a notion of non-crossing partitions and a notion of "clusters." The usual noncrossing partitions arise from the symmetric group, and clusters for the symmetric group are triangulations. For every finite Coxeter group W, the number of noncrossing partitions equals the number of clusters, but no natural connection has been known between clusters and noncrossing partitions. In this talk we define "diminishing elements" of W and connect diminishing elements both to noncrossing partitions and to clusters via natural bijections.