  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.
