The drop polynomial of a weighted digraph

Given a directed graph D=(V,E) with n vertices and a permutation π:V→V on its vertices, we say that π has a drop at a vertex u∈V is (u,π(u)) is an edge of D. Letting denote the number of permutations on V with precisely k drops, we can define the binomial drop polynomial . In this paper we study various properties of BD(x) and its generalization to the case when the edges of D are assigned weights. In particular, BD(x) satisfies a natural deletion/contraction recursion, quite similar to this type of recursion for the celebrated Tutte polynomial (for graphs) and the path/cycle cover polynomial for digraphs.