The Pop-Stack Operator on Ornamentation Lattices
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913654945873920 |
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| author | Ajran, Khalid Defant, Colin |
| author_facet | Ajran, Khalid Defant, Colin |
| contents | Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator $\mathsf{Pop}\colon\mathcal{O}(\mathsf{T})\to\mathcal{O}(\mathsf{T})$, which sends each element $δ$ to the meet of the elements covered by or equal to $δ$. We compute the maximum size of a forward orbit of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Defant for Tamari lattices. We also characterize the image of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Hong for Tamari lattices. For each integer $k\geq 0$, we provide necessary conditions for an element of $\mathcal{O}(\mathsf{T})$ to be in the image of $\mathsf{Pop}^k$. This allows us to completely characterize the image of $\mathsf{Pop}^k$ on a Tamari lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10311 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Pop-Stack Operator on Ornamentation Lattices Ajran, Khalid Defant, Colin Combinatorics 05A15, 06A07 Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator $\mathsf{Pop}\colon\mathcal{O}(\mathsf{T})\to\mathcal{O}(\mathsf{T})$, which sends each element $δ$ to the meet of the elements covered by or equal to $δ$. We compute the maximum size of a forward orbit of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Defant for Tamari lattices. We also characterize the image of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Hong for Tamari lattices. For each integer $k\geq 0$, we provide necessary conditions for an element of $\mathcal{O}(\mathsf{T})$ to be in the image of $\mathsf{Pop}^k$. This allows us to completely characterize the image of $\mathsf{Pop}^k$ on a Tamari lattice. |
| title | The Pop-Stack Operator on Ornamentation Lattices |
| topic | Combinatorics 05A15, 06A07 |
| url | https://arxiv.org/abs/2501.10311 |