  Commutative properties in formal languages pose problems at the frontier of
computer science, computational linguistics and computational group theory. A
prominent problem of this kind is the position of the language $O_n$, the
language that contains the same number of letters $a_i$ and $\bar a_i$ with
$1\leq i\leq n$, in the known classes of formal languages. It has recently been
shown that $O_n$ is a Multiple Context-Free Language (MCFL). However the more
precise conjecture of Nederhof that $O_n$ is an MCFL of dimension $n$ was left
open. We present two proofs of this conjecture, both relying on tools from
algebraic topology. On our way, we prove a variant of the necklace splitting
theorem.
