Order-preserving maps from a poset to a chain, the order polytope, and the Todd class of the associated toric variety

We give a new formula for the number of order-preserving maps from a finite poset Q to a finite chain, which sums a contribution from each face of the order polytope of Q. The formula we derive could also be obtained if we knew that a certain cycle represented the Todd class of the toric variety associated to Q. We conjecture that, for toric varieties arising from posets, this cycle does represent the Todd class; we prove this conjecture in the case where Q is a chain.