Recently I did some explicit computations that involved the BCH series, $\log(e^x e^y)$. Here $x$ and $y$ are non-commuting variables, and the BCH series lives in the graded completion $FL(x,y)$ of the free Lie algebra generated by $x$ and $y$.

Mostly by chance I found that when BCH is written in the Lyndon basis of $FL(x,y)$, the number of Lyndon words that occur in its degree $n$ piece is {2, 1, 2, 1, 6, 5, 18, 17, 55, 55, 186, 185, 630, 629, 2181, 2181, 7710, 7709, 27594, 27593, 99857, 99857}, for $n$ running from 1 to 22.

There is an obvious pattern in this sequence - it seems that the odd-numbered terms are almost equal to the even-numbered terms that follow them, with a decline of one in 2/3 of the times, and with precise equality in the remaining 1/3 of the times. I have no idea why this is so. Perhaps you do?

Why care? The truth is that I'm curious but I don't care much; I just stumbled upon this by chance. Yet Lyndon words are a very effective tool for computations in free Lie algebras, and the BCH formula appears in many of these computations. The fact that there is some unexpected symmetry in the Lyndon word description of BCH suggests that BCH contains less information than one might think, possibly leading to some computational advantage. Though in (my) reality, the computational bottlenecks are anyway elsewhere.

Some further details and observations are at http://drorbn.net/AcademicPensieve/2012-12/nb/BCH-Lyndon_Question.pdf.