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 $r: 2^E\to \mathbb{Z}{\geq 0}$ satisfying $r(S)\leq \min(r(E),|S|)$ for every set $S\subseteq E$, the Tutte polynomial $$T_M(x,y):=\sum{S\subseteq E}(x-1)^{r(E)-r(S)}(y-1)^{|S|-r(S)},$$ written as $T_M(x,y)=\sum_{i,j}t_{ij}x^iy^j$, satisfies that for any integer $h \geq 0$, we have $$\sum_{i=0}^h\sum_{j=0}^{h-i}\binom{h-i}{j}(-1)^jt_{ij}=(-1)^{m-r}\binom{h-r}{h-m},$$ where $r=r(E)$, and we use the convention that when $h<m$, the binomial coefficient $\binom{h-r}{h-m}$ is interpreted as $0$. This generalizes a theorem of Brylawski on matroid rank functions and $h<m$, and a theorem of Gordon for $h\leq m$ 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 $T_M(x,y)$ simplifies to $(x-1)^{r(E)}y^{|E|}$ along the hyperbola $(x-1)(y-1)=1$.