$Q_0$, and hence on the simple roots. Namely we require that $e_{ha}$ precedes $e_{ta}$ for every arrow $a\in Q_1$. We call this the *“head before tail” order* on simple roots. We describe an order on the positive roots $\Phi_+$ by forcing the condition that $$\label{eqn:pos.rts.order}
    \beta' \text{~precedes~} \beta'' \implies \langle \beta',\beta'' \rangle \geq 0.$$ The above condition on positive roots defines a partial order, from which we adopt any linear extension as a total order. The choice is not unique, but always exists for Dynkin quivers (see [@mr2010; @rr2013; @jarr2018; @ja2018] for more details and equivalent formulations). Because this order was first utilized in Reineke’s work [@mr2010] to prove Donaldson–Thomas type identities (a connection we will exploit later), we call the resulting total order on $\Phi_+$ a *Reineke order*; this terminology is also consistent with [@rr2018].

\[ex:Reineke.order.A3\] Consider the equioriented $A_3$ quiver $1 \leftarrow 2 \leftarrow 3$. Note that we have already chosen the names of the vertices so that the simple roots are in “head before tail” order $e_1\prec e_2 \prec e_3$. A Reineke order for the associated positive roots is given by $$\label{eqn:reineke.order.A3}
    \begin{aligned}
    \beta_1 & = e_3, & \beta_2 & = e_2+e_3, & \beta_3 & = e_2, 