Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915391519850496 |
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| author | Wang, Lintong Yan, Sherry H. F. |
| author_facet | Wang, Lintong Yan, Sherry H. F. |
| contents | A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein-Han-Kitaev-Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein-Han-Kitaev-Zhang concerning the shape-Wilf-equivalence for POPs of length $k$. This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns Wang, Lintong Yan, Sherry H. F. Combinatorics A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein-Han-Kitaev-Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein-Han-Kitaev-Zhang concerning the shape-Wilf-equivalence for POPs of length $k$. This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram. |
| title | Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.22098 |