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The coefficients of the chain polynomial of a finite poset enumerate chainsin the poset by their number of elements. The chain polynomials of thepartition lattices and their standard type $B$ analogues are shown to have onlyreal roots. The real-rootedness of the chain polynomial is conjectured for allgeometric lattices and is shown to be preserved by the pyramid and the prismoperations on Cohen--Macaulay posets. As a result, new families of convexpolytopes whose face lattices have real-rooted chain polynomials are presented.An application to the face enumeration of the second barycentric subdivision ofthe boundary complex of the simplex is also included.

Chain enumeration, partition lattices and polynomials with only real roots