We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number of blocks, we introduce and investigate “Stirling posets.” As we show, the Stirling posets have a hierarchy and they glue together to give the whole set partition poset. Moreover, we show that they (Stirling posets) are graded and EL-shellable. We offer various reformulations of their length functions and determine the recurrences for their length generating series.

中文翻译：

---

斯特林姿势

我们在集合分区上组合地定义偏序，并证明它等价于上三角矩阵上的 Bruhat-Chevalley-Renner 阶。通过考虑由具有固定数量块的集合分区组成的子集，我们引入并研究了“斯特林集”。正如我们所展示的，斯特林偏序集具有层次结构，它们粘合在一起以提供整个集合分区偏序集。此外，我们表明它们（斯特林偏序）是分级的和 EL-shellable。我们提供了它们的长度函数的各种重新表述，并确定了它们的长度生成序列的重复。