Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda}\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$.
We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda \u2080i>\lambda \u2081>...>{\lambda}_{n-1}$ and ordinals $\alpha \u2080,...,{\alpha}_{n-1}$ such that $\alpha =\alpha \u2080+\cdots +{\alpha}_{n-1}$ and $f=f\u2080\u23dcf\u2081\u23dc...\u23dc{f}_{n-1}$ where each ${f}_{i}{\in}_{{\lambda}_{i}}\left({\alpha}_{i}\right)$. Under GCH we prove that if α < ω₂ then
(i) ${}_{\omega}\left(\alpha \right)=s{\in}^{\alpha}\omega ,\omega \u2081:s\left(0\right)=\omega $;
(ii) if λ > cf(λ) = ω,
${}_{\lambda}\left(\alpha \right)=s{\in}^{\alpha}\lambda ,\lambda \u207a:s\left(0\right)=\lambda ,{s}^{-1}\lambda is\omega \u2081-closedin\alpha $;
(iii) if cf(λ) = ω₁,
${}_{\lambda}\left(\alpha \right)=s{\in}^{\alpha}\lambda ,\lambda \u207a:s\left(0\right)=\lambda ,{s}^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha $;
(iv) if cf(λ) > ω₁, ${}_{\lambda}\left(\alpha \right){=}^{\alpha}\lambda $.
This yields a complete characterization of the classes (α) for all α < ω₂,...