Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

Abstract

In this short note we show that a system M=(E,r) with a ground set E of size m and (rank) function r:2E→Z≥0 satisfying r(S)≤min(r(E),|S|) for every set S⊆E, the Tutte polynomial TM(x,y)≔∑S⊆E(x−1)r(E)−r(S)(y−1)|S|−r(S),written as TM(x,y)=∑i,jtijxiyj, satisfies that for any integer h≥0, we have ∑i=0h∑j=0h−ih−ij(−1)jtij=(−1)m−rh−rh−m,where r=r(E), and we use the convention that when h<m, the binomial coefficient h−rh−m is interpreted as 0.This generalizes a theorem of Brylawski on matroid rank functions and h<m, and a theorem of Gordon for h≤m with the same assumptions on the rank function.The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial TM(x,y) simplifies to (x−1)r(E)y|E| along the hyperbola (x−1)(y−1)=1.