• 1 Laboratoire de combinatoire et d'informatique mathématique [Montréal]

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 $\textit{graded}$ $(3+1)$-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.