Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.16793 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917996230868992 |
|---|---|
| author | Oger, Francis |
| author_facet | Oger, Francis |
| contents | For each $n\in N ^{\ast }$, we write $s_{n}=\left( 1,\ldots ,1,0\right) $ with $n$ times $1$. For each $a \in N$, we consider the binary representation $\left( a_{i}\right) _{i\in -N }$ of $a$ with $a_{i}=0$ for nearly each $i$; we denote by $α_{n}(a)$ the number of integers $i$ such that $\left( a_{i}, \ldots ,a_{i+n} \right) =s_{n}$. We consider the curve $C_{n}=\left( S_{n,k}\right) _{k\in N ^{\ast }}$ which consists of consecutive segments of length $1$ such that, for each $k$, $S_{n,k+1}$ is obtained from $S_{n,k}$ by turning right if $k+α_{n}(k)-α_{n}(k-1)$ is even and left otherwise. $C_{1}$ is self-avoiding since it is the curve associated to the alternating folding sequence. In [1], M. Mendès France and J. Shallit conjectured that the curves $C_{n}$ for $n\geq 2$ are also self-avoiding. In the present paper, we show that this property is true for $n=2$. We also prove that $C_{2}$ has some properties similar to those which were shown in [2], [3] and [4] for folding curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A self-avoiding curve associated with sums of digits Oger, Francis Combinatorics 05B45 (primary) 52C20 For each $n\in N ^{\ast }$, we write $s_{n}=\left( 1,\ldots ,1,0\right) $ with $n$ times $1$. For each $a \in N$, we consider the binary representation $\left( a_{i}\right) _{i\in -N }$ of $a$ with $a_{i}=0$ for nearly each $i$; we denote by $α_{n}(a)$ the number of integers $i$ such that $\left( a_{i}, \ldots ,a_{i+n} \right) =s_{n}$. We consider the curve $C_{n}=\left( S_{n,k}\right) _{k\in N ^{\ast }}$ which consists of consecutive segments of length $1$ such that, for each $k$, $S_{n,k+1}$ is obtained from $S_{n,k}$ by turning right if $k+α_{n}(k)-α_{n}(k-1)$ is even and left otherwise. $C_{1}$ is self-avoiding since it is the curve associated to the alternating folding sequence. In [1], M. Mendès France and J. Shallit conjectured that the curves $C_{n}$ for $n\geq 2$ are also self-avoiding. In the present paper, we show that this property is true for $n=2$. We also prove that $C_{2}$ has some properties similar to those which were shown in [2], [3] and [4] for folding curves. |
| title | A self-avoiding curve associated with sums of digits |
| topic | Combinatorics 05B45 (primary) 52C20 |
| url | https://arxiv.org/abs/2504.16793 |