Pattern avoidance and the fundamental bijection
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911948755435520 |
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| author | Archer, Kassie Laudone, Robert P. |
| author_facet | Archer, Kassie Laudone, Robert P. |
| contents | The fundamental bijection is a bijection $θ:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $π\in \mathcal{S}_n$ that avoids a pattern $σ\in \mathcal{S}_3$, whose image $θ(π)$ also avoids $σ$. We additionally consider what happens under repeated iterations of $θ$; in particular, we enumerate permutations $π\in \mathcal{S}_n$ that have the property that $π$ and its first $k$ iterations under $θ$ all avoid a pattern $σ$. Finally, we consider permutations with the property that $π=θ^2(π)$ that avoid a given pattern $σ$, and end the paper with some directions for future study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06338 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pattern avoidance and the fundamental bijection Archer, Kassie Laudone, Robert P. Combinatorics The fundamental bijection is a bijection $θ:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $π\in \mathcal{S}_n$ that avoids a pattern $σ\in \mathcal{S}_3$, whose image $θ(π)$ also avoids $σ$. We additionally consider what happens under repeated iterations of $θ$; in particular, we enumerate permutations $π\in \mathcal{S}_n$ that have the property that $π$ and its first $k$ iterations under $θ$ all avoid a pattern $σ$. Finally, we consider permutations with the property that $π=θ^2(π)$ that avoid a given pattern $σ$, and end the paper with some directions for future study. |
| title | Pattern avoidance and the fundamental bijection |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.06338 |