Enumerating alternating-runs with sign in Type A and Type B Coxeter groups

We enumerate alternating runs in the alternating group $A_n\subseteq S_n$. This leads us to enumerate the bivariate peak and valley polynomial with sign taken into account. We prove an exact formula for this signed enumerator in $S_n$ and show that this polynomial depends on the value of $n\left(\mathrm{mod}4\right)$. If $R_n\left(t\right)$ is the polynomial enumerating alternating runs in $S_n$, Wilf showed that $(1+t)$ divides $R_n\left(t\right)$ and determined the exponent of $(1+t)$ that divides $R_n\left(t\right)$. Let $R_n^{+}\left(t\right)$ and $R_n^{-}\left(t\right)$ be the polynomials enumerating alternating runs in $A_n$ and $S_n-A_n$ respectively. By finding the exponent of $(1+t)$ that divides $R_n^{\pm }\left(t\right)$, we refine Wilfs result when $n\equiv 2,3\left(\mathrm{mod}4\right)$. When $n\equiv 0,1\left(\mathrm{mod}4\right)$, we show that the exponent is one short of what Wilf obtains.

As applications of our results, we get moment type identities involving the coefficients of $R_n^{\pm }\left(t\right)$, refinements to enumerating alternating and unimodal permutations in $A_n$, $S_n-A_n$ and Central Limit Theorems about alternating runs in $A_n$ and $S_n-A_n$. Similar results are obtained for type B Coxeter groups.