Séminaire Lotharingien de Combinatoire, 84B.64 (2020), 11 pp.
Mahir Bilen Can and Yonah Cherniavsky
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.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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