The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.

翻译：排列的区间偏序集是捕捉排列的所有区间及其包含关系的一种有效方法，由 Tenner 近期引入。本文探讨了排列区间偏序集的几何解释。我们建立了树状区间偏序集与带有非交叉对角线的凸多边形之间的一一对应关系，为这一纯组合概念提供了一个新颖的几何视角。此外，我们利用这一对应关系对区间偏序集进行了计数，并展示了其在分块简单排列中的应用。