A self-dual poset on objects counted by the Catalan numbers and a type-B analogue

Abstract

We introduce two partially ordered sets, P_n^A and P_n^B, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of P_n^A and P_n^B are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties of these permutation posets follow, including self-duality and the strong Sperner property. We also discuss posets Q_n^A and Q_n^B similarly associated with noncrossing partitions, defined by means of the excedance sets of suitable pattern-avoiding subsets of the symmetric and hyperoctahedral groups.