$n_1, n_2$ for which the equations $$n_1(2n_{2}- n_1) =\left(\sum_{i=1}^{i=k}m_i^2\right) -1 \qquad n_1 + 2n_2 =\left(\sum_{i=1}^{i=k}m_i\right) +1$$ admit a vector solution $(m_1,\ldots, m_{k})\in{\mathbb{N}}^{k}$ (as we will see later, those equations simply ensure that the exceptional classes in the $k$-fold blow-up of $M_{\mu}^{i}$ have nonnegative symplectic areas). He also obtained the following bounds for the stability number of $M_{\mu}^{0}$: $$2 \mu \leq  N_{{\rm stab}}(M_{\mu}^{0})\leq 8 \mu$$

Building on [@Bi; @MP], F. Schlenk [@Sh2] later computed the packing numbers $p_{k}(M_{\mu}^{i})$, $i=0,1$, for $k\leq 7$ (those can be found in Appendix A), and proved that $$\max(8,2 \mu+1) \leq  N_{{\rm stab}}(M_{\mu}^{1}) \leq
\begin{cases}
8 \mu+4 & \text{~if~} \frac{1}{2}\leq \mu \\
\frac{2\mu+1}{\mu^{2}} & \text{~if~} \mu<\frac{1}{2}
\end{cases}$$

The above results reduce, in principle, the computation of the packing numbers $p_{k}(M_{\mu}^{i})$, $k\geq 8$, and of the stability numbers $N_{{\rm stab}}(M_{\mu}^{i})$ to purely arithmetic problems. However, since their general solutions are not known, they do not yield explicit formulae in terms of the parameters $k$ and $\mu$.

Main results
------------

In this paper, we use a modified version of Li-Li’s reduction algorithm [@Li-Li-SymplecticGenus],[@Li-Li-Diff], to compute the packing numbers, the generalized Gromov widths, and the stability numbers of rational ruled symplectic 4-manifolds. We also identify the exceptional homology classes that 