Stirling Posets

TL;DR

This paper introduces Stirling posets, a new combinatorial hierarchy of set partitions that are graded, EL-shellable, and relate to the Bruhat-Chevalley-Renner order on matrices, providing new insights into their structure.

Contribution

It defines Stirling posets as subposets of set partitions with fixed block counts, establishing their properties and connections to matrix orders.

Findings

Stirling posets are graded and EL-shellable.

They have a hierarchical structure that forms the entire set partition poset.

Recurrences for their length generating series are derived.

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.