Séminaire Lotharingien de Combinatoire, 84B.64 (2020), 11 pp.

Mahir Bilen Can and Yonah Cherniavsky

The Bruhat-Chevalley-Renner Order on the Set Partitions

Abstract. 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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