The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type \(B\) analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen-Macaulay posets. As a result, new families of convex polytopes whose barycentric subdivisions have real-rooted \(f\)-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.

Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10

Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag \(h\)-vector, convex polytope, barycentric subdivision