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Given a context-free language $L$, define the language $p(L)$ as containing all permutations of strings in $L$ (i.e. all strings in $L$ such that the order of symbols is not important). Is $p(L)$ context-free?

I found two papers dealing with similar, but not identical, questions:

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    $\begingroup$ Have you tried applying the techniques shown in our reference question? $\endgroup$
    – Raphael
    Commented Apr 15, 2013 at 12:16

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Start with simple context-free (or even regular) languages, and see what happens. For instance determine $p(L)$ for $L = (ab)^*$ and $L=(abc)^*$.

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    $\begingroup$ A penny just dropped in my head, thanks. $\endgroup$
    – Raphael
    Commented Apr 15, 2013 at 14:45
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No, in general not. For example the MIX language, consisting of equal numbers of $\{a,b,c\}$ in any order is not context-free but is the permutation of the regular language $(abc)^*$.

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