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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.05113 |
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| _version_ | 1866909132259328000 |
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| author | Choi, Yunseo Gan, Katelyn Li, Andrew Zhu, Tiffany |
| author_facet | Choi, Yunseo Gan, Katelyn Li, Andrew Zhu, Tiffany |
| contents | Recently, Xia introduced a deterministic variation $ϕ_σ$ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition $p$ is sorted by $ϕ^{N(p)}_{aba}$, where $N(p)$ is the number of distinct alphabets in $p$. Xia then asked which set partitions $p$ are not sorted by $ϕ_{aba}^{N(p)-1}$. In this note, we prove that the minimal length of a set partition $p$ that is not sorted by $ϕ_{aba}^{N(p)-1}$ is $2N(p)$. Then we show that there is only one set partition of length $2N(p)$ and ${{N(p) + 1} \choose 2} + 2{N(p) \choose 2}$ set partitions of length $2N(p)+1$ that are not sorted by $ϕ_{aba}^{N(p)-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05113 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the set partitions that require maximum sorts through the $aba-$avoiding stack Choi, Yunseo Gan, Katelyn Li, Andrew Zhu, Tiffany Combinatorics Recently, Xia introduced a deterministic variation $ϕ_σ$ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition $p$ is sorted by $ϕ^{N(p)}_{aba}$, where $N(p)$ is the number of distinct alphabets in $p$. Xia then asked which set partitions $p$ are not sorted by $ϕ_{aba}^{N(p)-1}$. In this note, we prove that the minimal length of a set partition $p$ that is not sorted by $ϕ_{aba}^{N(p)-1}$ is $2N(p)$. Then we show that there is only one set partition of length $2N(p)$ and ${{N(p) + 1} \choose 2} + 2{N(p) \choose 2}$ set partitions of length $2N(p)+1$ that are not sorted by $ϕ_{aba}^{N(p)-1}$. |
| title | On the set partitions that require maximum sorts through the $aba-$avoiding stack |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.05113 |