Structure of spaces of rhombus tilings in the lexicograhic case
Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1,...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2005-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3400/pdf |
| _version_ | 1797270337191149568 |
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| author | Éric Rémila |
| author_facet | Éric Rémila |
| author_sort | Éric Rémila |
| collection | DOAJ |
| description | Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element. |
| first_indexed | 2024-04-25T02:02:40Z |
| format | Article |
| id | doaj.art-3220a6ab10404b5f8db6bc796a771ac2 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:40Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-3220a6ab10404b5f8db6bc796a771ac22024-03-07T14:41:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AE,...Proceedings10.46298/dmtcs.34003400Structure of spaces of rhombus tilings in the lexicograhic caseÉric Rémila0https://orcid.org/0000-0002-9265-9907Institut Universitaire de Technologie [Roanne]Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element.https://dmtcs.episciences.org/3400/pdfrhombus tilingflipconnectivity[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Éric Rémila Structure of spaces of rhombus tilings in the lexicograhic case Discrete Mathematics & Theoretical Computer Science rhombus tiling flip connectivity [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Structure of spaces of rhombus tilings in the lexicograhic case |
| title_full | Structure of spaces of rhombus tilings in the lexicograhic case |
| title_fullStr | Structure of spaces of rhombus tilings in the lexicograhic case |
| title_full_unstemmed | Structure of spaces of rhombus tilings in the lexicograhic case |
| title_short | Structure of spaces of rhombus tilings in the lexicograhic case |
| title_sort | structure of spaces of rhombus tilings in the lexicograhic case |
| topic | rhombus tiling flip connectivity [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3400/pdf |
| work_keys_str_mv | AT ericremila structureofspacesofrhombustilingsinthelexicograhiccase |