The drop polynomial of a weighted digraph

Article history: Received 22 February 2016 Available online 23 April 2017 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 〈 D k 〉 denote the number of permutations on V with precisely k drops, we can define the binomial drop polynomial BD(x) = ∑ k 〈 D k 〉( x+k n ) . 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. © 2017 Elsevier Inc. All rights reserved.