Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial a finite poset enumerate chains in by their number elements. polynomials partition lattices and standard type \(B\) analogues are shown to have only real roots. real-rootedness is conjectured for all geometric be preserved pyramid prism operations on Cohen-Macaulay posets. As result, new families convex polytopes whose barycentric subdivisions real-rooted \(f\)-polynomials presented. An application face enumeration second subdivision boundary complex simplex also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, lattice, flag \(h\)-vector, polytope,