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The height of the Lyndon tree

Abstract : 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 factorization of $L_n$ and of the factors produced by these factorization is the $\textit{Lyndon tree}$ of $L_n$. We prove that the height $H_n$ of $\mathfrak{L} (L_n)$ satisfies $\lim \limits_n \frac{H_n}{\mathsf{ln}n}=\Delta$, in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta ≃5.092\dots $). The convergence is the convergence in probability of random variables.
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Lucas Mercier, Philippe Chassaing. The height of the Lyndon tree. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.957-968, ⟨10.46298/dmtcs.2357⟩. ⟨hal-01229697⟩

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