A Note on the Binomial Drop Polynomial of a Poset

Suppose (P,-<) is a poset of size n and n: P-~ P is a permutation. We say that n has a drop at x if n(x)~x. Let fie(k) denote the number of n having k drops, 0 <~ k < n, and define the drop polynomial A p(2) by Further, define the incomparability graph I(P) to have vertex set P and edges 0" whenever i and j are incomparable in P, i.e., neither i-<j nor j< i holds. In this note we give a short proof that Ae(2) is equal to the chromatic polynomial of ](P).