The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths
In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2014-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2409/pdf |
| _version_ | 1797270307127427072 |
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| author | Myrto Kallipoliti Henri Mühle |
| author_facet | Myrto Kallipoliti Henri Mühle |
| author_sort | Myrto Kallipoliti |
| collection | DOAJ |
| description | In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group. |
| first_indexed | 2024-04-25T02:02:11Z |
| format | Article |
| id | doaj.art-c6ebd7895b1f4eda82309d30c70dc1a9 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:11Z |
| publishDate | 2014-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-c6ebd7895b1f4eda82309d30c70dc1a92024-03-07T14:53:18ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502014-01-01DMTCS Proceedings vol. AT,...Proceedings10.46298/dmtcs.24092409The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck PathsMyrto Kallipoliti0https://orcid.org/0000-0003-2188-6552Henri Mühle1Fakultät für Mathematik [Wien]Fakultät für Mathematik [Wien]In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group.https://dmtcs.episciences.org/2409/pdfm-tamari latticem-dyck pathsm-cover posetfuss-catalan combinatoricssymmetric groupdihedral groupleft-modularitytrimnessmöbius function[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Myrto Kallipoliti Henri Mühle The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths Discrete Mathematics & Theoretical Computer Science m-tamari lattice m-dyck paths m-cover poset fuss-catalan combinatorics symmetric group dihedral group left-modularity trimness möbius function [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths |
| title_full | The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths |
| title_fullStr | The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths |
| title_full_unstemmed | The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths |
| title_short | The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths |
| title_sort | m cover posets and the strip decomposition of m dyck paths |
| topic | m-tamari lattice m-dyck paths m-cover poset fuss-catalan combinatorics symmetric group dihedral group left-modularity trimness möbius function [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/2409/pdf |
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