The height of the Lyndon tree
Résumé
We consider the set $\mathcal{L}_{n}$ of $n$-letters long Lyndon words on the alphabet $\mathcal A=\{0,1\}$. For a random uniform element $L_{n}$ of the set $\mathcal{L}_{n}$, the binary tree $\mathfrak{L}(L_{n})$ obtained by successive standard factorizations of $L_{n}$ and of the factors produced by these factorizations is the \emph{Lyndon tree} of $L_{n}$. We prove that the height $H_n$ of $\mathfrak{L}(L_{n})$ satisfies \[\lim_{n}\frac{H_{n}}{\ln n}=\Delta,\] in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta\simeq 5.092\dots$). The convergence is the convergence in probability of random variables.
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