Lucas Mercier ; Philippe Chassaing - The height of the Lyndon tree

dmtcs:2357 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2357
The height of the Lyndon tree

Authors: Lucas Mercier 1; Philippe Chassaing 1

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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Yule process, branching random walk, binary search tree, Galton-Watson tree, Lyndon tree,Lyndon word,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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Source : ScholeXplorer IsRelatedTo DOI 10.1016/0097-3165(90)90050-7
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