Non-crossing partitions, non-nesting partitions and Coxeter sortable elements in types A and B

Details Non-crossing partitions, non-nesting partitions and Coxeter sortable elements in types A and B Non-crossing partitions, non-nesting partitions and Coxeter sortable elements in types A and B

2008-08-20

arxiv.org/abs/0808.2822v1

Mathematics

Combinatorics

27 pages, 11 figures

Scientific paper

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic refection 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 partitions 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.

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