Éléments de comptage sur les générateurs du groupe modulaire et les $λ$-quiddités
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915134751899648 |
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| author | Mabilat, Flavien |
| author_facet | Mabilat, Flavien |
| contents | The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_{1} & -1 \\[4pt] 1 & 0 \end{pmatrix}=\pm M$ when $M$ is equal to the generators of the modular group $S=\begin{pmatrix} 0 & -1 \\[4pt] 1 & 0 \end{pmatrix}$ and $T=\begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}$. To count these elements, we will study the $λ$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Éléments de comptage sur les générateurs du groupe modulaire et les $λ$-quiddités Mabilat, Flavien Combinatorics The aim of this article is to count the $n$-tuples of positive integers $(a_{1},\ldots,a_{n})$ solutions of the equation $\begin{pmatrix} a_{n} & -1 \\[4pt] 1 & 0 \end{pmatrix} \begin{pmatrix} a_{n-1} & -1 \\[4pt] 1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_{1} & -1 \\[4pt] 1 & 0 \end{pmatrix}=\pm M$ when $M$ is equal to the generators of the modular group $S=\begin{pmatrix} 0 & -1 \\[4pt] 1 & 0 \end{pmatrix}$ and $T=\begin{pmatrix} 1 & 1 \\[4pt] 0 & 1 \end{pmatrix}$. To count these elements, we will study the $λ$-quiddities, which are the solutions of the equation in the case $M=Id$ (related to Coxeter's friezes), whose last component is fixed. |
| title | Éléments de comptage sur les générateurs du groupe modulaire et les $λ$-quiddités |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.01328 |