On log concavity for order-preserving maps of partial orders

Stanley used the Aleksandrov-Fenchel inequalities from the theory of mixed volumes to prove the following results. Let P be a partially ordered set with n elements, and let x ε{lunate} P, If N _i ^* is the number of linear extension λ : P → {1, 2, ..., n} satisfying λ(x) = i, then the sequence N ₁ ^*, ..., N _n ^* is log concave (and therefore unimodal). Here the analogous results for both strict order-preserving and order-preserving maps are proved using an explicit injection. Further, if ν _c is the number of strict order-preserving maps of P into a chain of length c, then ν _c is shown to be log concave, and the corresponding result is established for order-preserving maps.