The Bruhat–Chevalley–Renner Order on the Set Partitions

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.