Positive roots of type A
= Positive roots of type A
{title2=$\mathbf n=1_{[a,b]},\quad1\leq a\leq b\leq m$}
For the chain, with $n_0=n_{m+1}=0$, the quadratic form is $\tfrac12\sum_{i=0}^m(n_{i+1}-n_i)^2$. Its value is one precisely for interval vectors: one $+1$ difference followed by one $-1$, all other differences zero. Thus there are $m(m+1)/2$ positive roots, independently of orientation.