Permutations of context-free, ET0L and indexed languages

For a language L, we consider its cyclic closure, and more generally the language C(L), which consists of all words obtained by partitioning words from L into k factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators C. This both sharpens and generalises Brandstädt’s result that if L is context-free then C(L) is context-sensitive and not context-free in general for k ≥ 3. We also show that the cyclic closure of an indexed language is indexed.