For each positive integer k, we consider five well-studied posets defined on the set of Dyck paths of semilength k. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.

中文翻译：

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加泰罗尼亚语间隔和唯一排序的排列

对于每个正整数k，我们考虑在半长为k的Dyck路径集上定义了五个经过充分研究的波姿。我们证明了避免各种模式的唯一排序排列与这些位姿中的间隔相等。虽然我们的大多数证明都是双射的，但有些使用生成树和生成函数。我们以几个猜想结束。