A Characterization of (3+1)-Free Posets

Posets containing no subposet isomorphic to the disjoint sums of chains 3+1 and/or 2+2 are known to have many special properties. However, while posets free of 2+2 and posets free of both 2+2 and 3+1 may be characterized as interval orders, no such characterization is known for posets free of only 3+1. We give here a characterization of (3+1)-free posets in terms of their antiadjacency matrices. Using results about totally positive matrices, we show that this characterization leads to a simple proof that the chain polynomial of a (3+1)-free poset has only real zeros.