Rees products and lexicographic shellability

TL;DR

This paper explores how lexicographic shellability can be used to analyze Rees products of posets, linking their homology ranks to combinatorial objects like derangements, and generalizing previous results.

Contribution

It introduces new examples of Rees products whose homology ranks count combinatorial objects, extending Jonsson's work with broader applications.

Findings

Homology rank of Rees products enumerates combinatorial objects

Generalization of Jonsson's derangement result

Examples linking poset topology to combinatorics

Abstract

We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural statistic on the set. Many of these examples generalize a result of J. Jonsson, which says that the rank of the unique nontrivial homology group of the Rees product of a truncated Boolean algebra of degree $n$ and a chain of length $n-1$ is the number of derangements in $\S_n$.\