Annals of Combinatorics

, Volume 18, Issue 4, pp 645–674 | Cite as

Structure and Enumeration of (3+1)-Free Posets

  • Mathieu Guay-Paquet
  • Alejandro H. Morales
  • Eric Rowland
Article

Abstract

A poset is (3+1)-free if it does not contain the disjoint union of chains of lengths 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.

Keywords

(3+1)-free posets trace monoid generating functions chromatic symmetric function 

Mathematics Subject Classification

05A15 05A16 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Mathieu Guay-Paquet
    • 1
  • Alejandro H. Morales
    • 1
  • Eric Rowland
    • 1
  1. 1.LaCIMUniversité du Québec à MontréalMontréalCanada

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