Structure and enumeration of (3+1)-free posets (extended abstract)

TL;DR

This paper develops a method to enumerate all (3+1)-free posets by decomposing them into bipartite graphs, providing generating functions for both labelled and unlabelled cases, advancing understanding of their structure.

Contribution

It introduces a novel decomposition approach for (3+1)-free posets and derives their enumeration formulas, addressing an open problem in combinatorics.

Findings

Derived generating functions for labelled (3+1)-free posets

Derived generating functions for unlabelled (3+1)-free posets

Provided a structural decomposition into bipartite graphs

Abstract

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the (3+1)-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have enumerated \emph{graded} (3+1)-free posets, but until now the general enumeration problem has remained open. We enumerate all (3+1)-free posets by giving a decomposition into bipartite graphs, and obtain generating functions for (3+1)-free posets with labelled or unlabelled vertices.