Difference ascent sequences and related combinatorial structures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915111631847424 |
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| author | Zang, Yongchun Zhou, Robin D. P. |
| author_facet | Zang, Yongchun Zhou, Robin D. P. |
| contents | Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled $(2+2)$-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length $3$, and Stoimenow matchings. Analogous results for weak ascent sequences have been obtained by Bényi, Claesson and Dukes. Recently, Dukes and Sagan introduced a more general class of sequences which are called $d$-ascent sequences. They showed that some maps from the weak case can be extended to bijections for general $d$ while the extensions of others continue to be injective but not surjective. The main objective of this paper is to restore these injections to bijections. To be specific, we introduce a class of permutations which we call difference $d$ permutations and a class of factorial posets which we call difference $d$ posets, both of which are shown to be in bijection with $d$-ascent sequences. Moreover, we also give a direct bijection between a class of matrices with a certain column restriction and Fishburn matrices. Our results give answers to several questions posed by Dukes and Sagan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Difference ascent sequences and related combinatorial structures Zang, Yongchun Zhou, Robin D. P. Combinatorics 05A05, 05C30 Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev, and are in bijection with unlabeled $(2+2)$-free posets, Fishburn matrices, permutations avoiding a bivincular pattern of length $3$, and Stoimenow matchings. Analogous results for weak ascent sequences have been obtained by Bényi, Claesson and Dukes. Recently, Dukes and Sagan introduced a more general class of sequences which are called $d$-ascent sequences. They showed that some maps from the weak case can be extended to bijections for general $d$ while the extensions of others continue to be injective but not surjective. The main objective of this paper is to restore these injections to bijections. To be specific, we introduce a class of permutations which we call difference $d$ permutations and a class of factorial posets which we call difference $d$ posets, both of which are shown to be in bijection with $d$-ascent sequences. Moreover, we also give a direct bijection between a class of matrices with a certain column restriction and Fishburn matrices. Our results give answers to several questions posed by Dukes and Sagan. |
| title | Difference ascent sequences and related combinatorial structures |
| topic | Combinatorics 05A05, 05C30 |
| url | https://arxiv.org/abs/2405.03275 |