The Rees product of posets

TL;DR

This paper investigates how the flag f-vector and Möbius function of graded posets change under the Rees product with chains and t-ary trees, providing new combinatorial and homological insights.

Contribution

It characterizes the effect of Rees products on flag f-vectors and Möbius functions, and explores enumerative and homological properties of specific Rees product posets.

Findings

Möbius function of Rees product with chains equals that of the dual poset's Rees product.

Möbius function of Rees product of cubical lattice with chain expressed as n times a signed derangement number.

Explicit basis for reduced homology and representation over the symmetric group.

Abstract

We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the M\"obius function of the Rees product of any graded poset with the chain, and more generally, the t-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, t-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the M\"obius function of this poset can be expressed as n times a signed derangement number. From this we derive a new bijective proof of Jonsson's result that the M\"obius function of the Rees product of the Boolean algebra with the chain is given by a derangement number. Using poset homology techniques we find an explicit basis for the reduced homology and determine a representation for the reduced homology of the order complex of the Rees product of the cubical lattice with the chain over the symmetric group.