Arrow pattern avoidance in permutations: structure and enumeration
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917314338750464 |
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| author | Archer, Kassie Laudone, Robert P. |
| author_facet | Archer, Kassie Laudone, Robert P. |
| contents | Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04218 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arrow pattern avoidance in permutations: structure and enumeration Archer, Kassie Laudone, Robert P. Combinatorics 05A05 Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation. |
| title | Arrow pattern avoidance in permutations: structure and enumeration |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2603.04218 |