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

TL;DR

This paper develops a comprehensive decomposition method for (3+1)-free posets, enabling enumeration, automorphism group analysis, and asymptotic counting for these structures.

Contribution

It introduces a new bipartite graph decomposition applicable to all (3+1)-free posets, advancing enumeration and structural understanding.

Findings

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

Provided automorphism group decompositions

Established asymptotic counts for (3+1)-free posets

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 play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.