# A characterization of 3+1-free posets

## ABSTRACT

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

Speaker's Contact Info: skan(at-sign)math.mit.edu