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.
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MGP was supported by an NSERC Postdoctoral Fellowship.
AHM was supported by a CRM-ISM Postdoctoral Fellowship.
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Guay-Paquet, M., Morales, A.H. & Rowland, E. Structure and Enumeration of (3+1)-Free Posets. Ann. Comb. 18, 645–674 (2014). https://doi.org/10.1007/s00026-014-0249-2
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DOI: https://doi.org/10.1007/s00026-014-0249-2