Doubling modulo odd integers, generalizations, and unexpected occurrences
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908336580984832 |
|---|---|
| author | Allouche, Jean-Paul Stipulanti, Manon Yao, Jia-Yan |
| author_facet | Allouche, Jean-Paul Stipulanti, Manon Yao, Jia-Yan |
| contents | The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive integer $n$. More precisely, this doubling map defines a permutation $σ_{2,n}$ and each of $A$ and $B$ counts the number $C_2(n)$ of cycles of $σ_{2,n}$, hence $A=B$. In the first part of this note, we give a direct proof of this last equality. To do so, we consider and study a generalized $(k,n)$-perfect shuffle permutation $σ_{k,n}$, where we multiply by an integer $k\ge 2$ instead of $2$, and its number $C_k(n)$ of cycles. The second part of this note lists some of the many occurrences and applications of the doubling map and its generalizations in the literature: in mathematics (combinatorics of words, dynamical systems, number theory, correcting algorithms), but also in card-shuffling, juggling, bell-ringing, poetry, and music composition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Doubling modulo odd integers, generalizations, and unexpected occurrences Allouche, Jean-Paul Stipulanti, Manon Yao, Jia-Yan Number Theory 11B50, 11B83, (primary), 05A05, 05A19, 11A25, 20B30, 20B99, 00A65 (secondary) The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive integer $n$. More precisely, this doubling map defines a permutation $σ_{2,n}$ and each of $A$ and $B$ counts the number $C_2(n)$ of cycles of $σ_{2,n}$, hence $A=B$. In the first part of this note, we give a direct proof of this last equality. To do so, we consider and study a generalized $(k,n)$-perfect shuffle permutation $σ_{k,n}$, where we multiply by an integer $k\ge 2$ instead of $2$, and its number $C_k(n)$ of cycles. The second part of this note lists some of the many occurrences and applications of the doubling map and its generalizations in the literature: in mathematics (combinatorics of words, dynamical systems, number theory, correcting algorithms), but also in card-shuffling, juggling, bell-ringing, poetry, and music composition. |
| title | Doubling modulo odd integers, generalizations, and unexpected occurrences |
| topic | Number Theory 11B50, 11B83, (primary), 05A05, 05A19, 11A25, 20B30, 20B99, 00A65 (secondary) |
| url | https://arxiv.org/abs/2504.17564 |