First, remember that you don't need numerical prefixes (e.g. di, tri, tetra, etc.) when naming ionic compounds. So, Al2(MnO4)3 is aluminum permanganate manganate. (note: since the aluminum ion has a charge of +3, this means that the anion has a -2 charge. Therfore, the anion is manganate [MnO42-], not permanganate [MnO4-])
These formulas are tricky since they involve very weird ions. For example, mercury has two common oxidation states. Mercury (II) is Hg2+, but mercury (I) is Hg22+. I don't know why the mercury (I) ions form a dimer, but that's how it behaves. Similarly, oxygen normally will form oxide ions (O2-), but it can also form peroxide ions (O2-).
Unfortunately, I don't know the name of the N2O2- ion either.
[edit: it's actually aluminum manganate, not aluminum permanganate]