Weakly Consecutive Sequences
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917569867284480 |
|---|---|
| author | Garrison, Thomas Seiler, Chris Knowles, Andrew |
| author_facet | Garrison, Thomas Seiler, Chris Knowles, Andrew |
| contents | A weakly consecutive sequence (WCS) is a permutation $σ$ of $\{1, \ldots, k\}$ such that if an integer $d$ divides $σ(i)$, then $d$ also divides $σ(i \pm d)$ insofar as these are defined. The structure of weakly consecutive sequences is surprisingly rich, and it is difficult to find a formula for the number $N(k)$ of WCS's of length $k$. However, for a given $k$ we describe four starting sequences, to each of which we can apply three \emph{rules} or operations to generate new WCS's. We conjecture that any WCS can be constructed by applying these rules, which depend in an intricate way on the primality of $k$ and surrounding integers. We find bounds for $N(k)$ by analyzing these rules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weakly Consecutive Sequences Garrison, Thomas Seiler, Chris Knowles, Andrew Combinatorics A weakly consecutive sequence (WCS) is a permutation $σ$ of $\{1, \ldots, k\}$ such that if an integer $d$ divides $σ(i)$, then $d$ also divides $σ(i \pm d)$ insofar as these are defined. The structure of weakly consecutive sequences is surprisingly rich, and it is difficult to find a formula for the number $N(k)$ of WCS's of length $k$. However, for a given $k$ we describe four starting sequences, to each of which we can apply three \emph{rules} or operations to generate new WCS's. We conjecture that any WCS can be constructed by applying these rules, which depend in an intricate way on the primality of $k$ and surrounding integers. We find bounds for $N(k)$ by analyzing these rules. |
| title | Weakly Consecutive Sequences |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.09497 |