  The alternating-runs polynomial enumerates alternating runs in the symmetric
group. There are three formulae for the number of permutations, $R_{n,k}$ in
$\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We
show that when enumerated with sign taken into account, one gets a {\it neat
formula}. As a consequence, we get a near refinement of a result of Wilf on the
exponent of $(1+t)$ when it divides the alternating-runs polynomial in the
alternating group $\mathcal{A}_n$. Other applications include a moment-type
identity and enumeration of alternating permutations in $\mathcal{A}_n$.
Similar results are obtained for the type B and type D Coxeter groups.
