Positive roots of type A ($\mathbf{n}=1_{[a,b]},1\le a\le b\le m$)

For the chain, with $n_0=n_{m+1}=0$, the quadratic form is $\frac{1}{2}{\sum }_{i=0}^m(n_{i+1}-n_i{)}^2$. 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.