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The homogeneous component of degree $k$ in the free Lie algebra $\mathfrak{Lie}(x_1,\dots,x_n)$ in $n$ letters is of dimension $$g_n(k)=\frac{1}{k}\sum_{d|k}\mu(d)n^{k/d}.$$ This is also the number of Lyndon words of length $k$ in $n$ letters, and of a few other things...

Question: Is there a positive formula for this number?

As an aside,

Question: Is there a corresponding formula for the dimensions of the homogeneous components of the free Lie triple system on $n$ letters?

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Perhaps you meant the mu function instead of the phi function and n instead of m? In any case, I don't see a reason to expect a positive formula here; look at the case when n is prime. –  Qiaochu Yuan May 6 '10 at 20:44
Er, whoops; I meant the case where k is prime. You get (n^k - n)/k and there's really no sensible way around that minus sign that I can see. –  Qiaochu Yuan May 6 '10 at 20:52
Andy- positive formulae are like pornography, I know them when I see them. In general it's a sum where all the terms are positive and hopefully in bijection with known sets. –  Ben Webster May 6 '10 at 22:45
@Ben: your comment really confuses me... since for a basis of the free Lie algebra we can take Lyndon words (a very combinatorial object), the above formula can be thought of as a sum of one term which has a clear combinatorial meaning! –  Vladimir Dotsenko May 7 '10 at 5:27
I second Vladimir's comment. I'm also confused: as Mariano mentions, Lyndon words (or any Hall set in the free magma, for that matter, as in chapter II of Bourbaki's Lie groups and Lie algebras) are a basis for the free Lie algebra. Why isn't this a "positive formula"? I guess the point must really be the second question about triple systems? –  GS May 11 '10 at 10:59

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This last bit is a bit mysterious to me.

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