 Girth determines initial chromatic coefficients

For a finite simple graph on $n$ vertices with $m$ edges and no graph cycle shorter than $\ell$, the coefficient of $x^{n-i}$ in its chromatic polynomial is $(-1{)}^i(\genfrac{}{}{0ex}{}{m}{i})$ for $0\le i\le \min (n,\ell -2)$. In the chromatic polynomial inclusion-exclusion formula, an edge subset containing a graph cycle of length at least $\ell$ has $n-c(F)\ge \ell -1$. Thus every contribution to $x^{n-i}$ in this range is a forest, with exactly $i$ edges. Conversely, all $i$-edge subsets are forests, so precisely $\left(\genfrac{}{}{0ex}{}{m}{i}\right)$ terms contribute, each with sign $(-1{)}^i$.