Dissertation/ Thesis
Combinatorics of permutreehedra and geometry of s-permutahedra ; Combinatoire des permusylvèdres et géométrie des s-permutaèdres
| العنوان: | Combinatorics of permutreehedra and geometry of s-permutahedra ; Combinatoire des permusylvèdres et géométrie des s-permutaèdres |
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| المؤلفون: | Tamayo Jiménez, Daniel |
| المساهمون: | Laboratoire Interdisciplinaire des Sciences du Numérique (LISN), Institut National de Recherche en Informatique et en Automatique (Inria)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Université Paris-Saclay, Viviane Pons, Vincent Pilaud |
| المصدر: | https://theses.hal.science/tel-04302406 ; Combinatorics [math.CO]. Université Paris-Saclay, 2023. English. ⟨NNT : 2023UPASG066⟩. |
| بيانات النشر: | CCSD |
| سنة النشر: | 2023 |
| مصطلحات موضوعية: | Partial orders, Algebraic combinatorics, Discrete geometry, Polytopes, Lattice quotients, Permutations, Ordres partiels, Combinatoire algébrique, Géométrie discrète, Quotients de treillis, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] |
| الوصف: | In algebraic combinatorics, lattices are partially ordered sets which possess both meet and join operations. The weak order on permutations is a classical example of a lattice that has a rich combinatorial structure. This has made it a starting point from which other combinatorial objects have been defined. For this thesis, we focus on studying two different families of lattices in relation to the weak order: the permutree lattices and the s-weak order. The first part of the thesis involves the theory of lattice quotients of the weak order building upon the work of N. Reading, specifically focusing on the family of permutree quotients of the weak order. Considering them as permutrees, as done by V. Pilaud and V. Pons, we extend the technology of bracket vectors from binary trees by defining inversion and cubic vectors. The inversion vector captures the meet operation of these lattices while the cubic vector helps realizes them geometrically via a cubical configuration. Changing our point of view and studying these quotients through the minimal elements of their congruence classes, we use the Coxeter Type A description of permutations to characterize permutrees using automata. These automata capture the pattern avoidance of ijk and/or kij implied by these quotients and allow us to define algorithms which generalize stack sorting. In the case where the quotient corresponds to a Cambrian lattice we relate our automata with Coxeter sorting. We give some insight about the same phenomenon for Coxeter groups of types B and D. The second part of this thesis stems from the work of V. Pons and C. Ceballos who defined the s-weak order on s-decreasing trees where s is a sequence of non-negative integers. In the case of s=(1,ldots,1) this definition recovers the weak order. In their first article, the authors conjectured that the s-permutahedron could be realized in space as a polyhedral subdivision of a zonotope. We give a positive answer to their conjecture when s is a sequence of positive integers by defining a graph ... |
| نوع الوثيقة: | doctoral or postdoctoral thesis |
| اللغة: | English |
| Relation: | NNT: 2023UPASG066 |
| الاتاحة: | https://theses.hal.science/tel-04302406 https://theses.hal.science/tel-04302406v1/document https://theses.hal.science/tel-04302406v1/file/114473_TAMAYO_2023_diffusion.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.66ADF00E |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |