Non-crossing Partitions, Non-nesting Partitions and Coxeter Sortable Elements in Types a and B

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic reflection groups. In particular, we explore Dyck paths of type B together with an area statistic and a major index. Then, we construct bijections between non-nesting and reverse non-crossing partitions for types A and B. These bijections simultaneously send the area statistic and the major index on non-nesting partitions to the length function and to the sum of the major index and the inverse major index on reverse non-crossing partitions. Finally, we construct bijections between non-nesting partitions and Coxeter sortable elements for types A and B having exactly the same properties.