Enumeration of chains in partially ordered sets and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset
enumerate chains in the poset by their number of elements. The chain
polynomials of geometric lattices are conjectured in this talk to have
only real roots and the analogous question is discussed for other
important classes of Cohen--Macaulay posets. Among other partial
results, the conjecture is proven for the subspace and the partition
lattices. An application to the face enumeration of the second
barycentric subdivision of the boundary complex of the simplex will
be given. This is joint work with Katerina Kalampogia-Evangelinou.