I have 3 more questions about maximal words (which are just another way of talking of necklaces).

Let W be a finite word on a two symbol alphabet {0,1}; let us say that W is maximal if it is the last item in the list of all its cyclic permutation (ordered lexicographically).

The number w(n) of maximal words of length n can be expressed with the aid of Eulers' totient function (see Minimal words of length n).

**Question 1**: What can be said about the asymptotics of w(n)? I expect that $lim (1/n) \log w(n) = h$ for some positive value h ...

**Question 2:** **The bisection scheme.**
Let us play the following game: start with the two string list [1,0] (which are both maximal) then we put in between the string obtained concatenating them, so we obtain the list [1,10,0], we go on like this (for any two neighbouring strings S,T we put the string ST in between), obtaining in turn the lists [1,110,10,100,0], [1,1110,110,11010,10,10100,100,1000,0] ... und so weiter.

I guess (and almost can prove) that in this way you generate all (and only) **primitive maximal words** of any length (let us say that a maximal word is *primitive* if it is not the repetition of a shorter one).

Has anybody a nice proof of this?

**Question 3:** is there an asymptotic distribution for the lengths of primitive maximal words generated by n runs of the prvious algorithm?