Relevant bibliographies by topics / Lattices and Combinatorics

Academic literature on the topic 'Lattices and Combinatorics'

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Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Lattices and Combinatorics"

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Mühle, Henri. "Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group." Annals of Combinatorics 25, no. 2 (April 10, 2021): 307–44. http://dx.doi.org/10.1007/s00026-021-00532-9.

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AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.
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Clingher, Adrian, and Jae-Hyouk Lee. "Lorentzian Lattices and E-Polytopes." Symmetry 10, no. 10 (September 28, 2018): 443. http://dx.doi.org/10.3390/sym10100443.

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We consider certain E n -type root lattices embedded within the standard Lorentzian lattice Z n + 1 ( 3 ≤ n ≤ 8 ) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice Z n + 1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type ( n − 4 ) 21 .
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Mühle, Henri. "Hochschild lattices and shuffle lattices." European Journal of Combinatorics 103 (June 2022): 103521. http://dx.doi.org/10.1016/j.ejc.2022.103521.

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Baranovskii, Evgenii, and Viatcheslav Grishukhin. "Non-rigidity Degree of a Lattice and Rigid Lattices." European Journal of Combinatorics 22, no. 7 (October 2001): 921–35. http://dx.doi.org/10.1006/eujc.2001.0510.

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Cuntz, Michael, Sophia Elia, and Jean-Philippe Labbé. "Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids." Annals of Combinatorics 26, no. 1 (November 8, 2021): 1–85. http://dx.doi.org/10.1007/s00026-021-00555-2.

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AbstractA catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.
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Elekes, György. "On Linear Combinatorics III. Few Directions and Distorted Lattices." Combinatorica 19, no. 1 (January 1, 1999): 43–53. http://dx.doi.org/10.1007/s004930050044.

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Vaderlind, Paul. "Clutters and Atomistic Lattices." European Journal of Combinatorics 7, no. 4 (October 1986): 389–96. http://dx.doi.org/10.1016/s0195-6698(86)80010-5.

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Nimbhorkar, Shriram K., and Deepali B. Banswal. "Generalizations of supplemented lattices." AKCE International Journal of Graphs and Combinatorics 16, no. 1 (April 1, 2019): 8–17. http://dx.doi.org/10.1016/j.akcej.2018.02.005.

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Leoreanu-Fotea, Violeta, and Ivo G. Rosenberg. "Hypergroupoids determined by lattices." European Journal of Combinatorics 31, no. 3 (April 2010): 925–31. http://dx.doi.org/10.1016/j.ejc.2009.06.005.

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Deza, Antoine, and Shmuel Onn. "Solitaire Lattices." Graphs and Combinatorics 18, no. 2 (May 1, 2002): 227–43. http://dx.doi.org/10.1007/s003730200016.

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Dissertations / Theses on the topic "Lattices and Combinatorics"

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Xin, Yuxin. "Strongly Eutactic Lattices From Vertex Transitive Graphs." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/cmc_theses/2171.

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In this thesis, we provide an algorithm for constructing strongly eutactic lattices from vertex transitive graphs. We show that such construction produces infinitely many strongly eutactic lattices in arbitrarily large dimensions. We demonstrate our algorithm on the example of the famous Petersen graph using Maple computer algebra system. We also discuss some additional examples of strongly eutactic lattices obtained from notable vertex transitive graphs. Further, we study the properties of the lattices generated by product graphs, complement graphs, and line graphs of vertex transitive graphs. This thesis is based on the research paper written by the author jointly with L. Fukshansky, D. Needell and J. Park.
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Goodwin, Michelle. "Lattices and Their Application: A Senior Thesis." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/cmc_theses/1317.

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Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick's Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick's Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick's Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research.
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Usatine, Jeremy. "Arithmetical Graphs, Riemann-Roch Structure for Lattices, and the Frobenius Number Problem." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/hmc_theses/57.

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If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine. We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.
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Alexander, Matthew R. "Combinatorial and Discrete Problems in Convex Geometry." Kent State University / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=kent1508949236617778.

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Krohne, Edward. "Continuous Combinatorics of a Lattice Graph in the Cantor Space." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc849680/.

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We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.
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Heuer, Manuela. "Combinatorial aspects of root lattices and words." Thesis, Open University, 2010. http://oro.open.ac.uk/24046/.

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This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly.
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Yoon, Young-jin. "Characterizations of Some Combinatorial Geometries." Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc277894/.

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We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.
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Melczer, Stephen. "Analytic Combinatorics in Several Variables : Effective Asymptotics and Lattice Path Enumeration." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSEN013/document.

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La combinatoire analytique étudie le comportement asymptotique des suites à travers les propriétés analytiques de leurs fonctions génératrices. Ce domaine a conduit au développement d’outils profonds et puissants avec de nombreuses applications. Au delà de la théorie univariée désormais classique, des travaux récents en combinatoire analytique en plusieurs variables (ACSV) ont montré comment calculer le comportement asymptotique d’une grande classe de fonctions différentiellement finies:les diagonales de fractions rationnelles. Cette thèse examine les méthodes de l’ACSV du point de vue du calcul formel, développe des algorithmes rigoureux et donne les premiers résultats de complexité dans ce domaine sous des hypothèses très faibles. En outre, cette thèse donne plusieurs nouvelles applications de l’ACSV à l’énumération des marches sur des réseaux restreintes à certaines régions: elle apporte la preuve de plusieurs conjectures ouvertes sur les comportements asymptotiques de telles marches,et une étude détaillée de modèles de marche sur des réseaux avec des étapes pondérées
The field of analytic combinatorics, which studies the asymptotic behaviour ofsequences through analytic properties of their generating functions, has led to thedevelopment of deep and powerful tools with applications across mathematics and thenatural sciences. In addition to the now classical univariate theory, recent work in thestudy of analytic combinatorics in several variables (ACSV) has shown how to deriveasymptotics for the coefficients of certain D-finite functions represented by diagonals ofmultivariate rational functions. This thesis examines the methods of ACSV from acomputer algebra viewpoint, developing rigorous algorithms and giving the firstcomplexity results in this area under conditions which are broadly satisfied.Furthermore, this thesis gives several new applications of ACSV to the enumeration oflattice walks restricted to certain regions. In addition to proving several openconjectures on the asymptotics of such walks, a detailed study of lattice walk modelswith weighted steps is undertaken
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Davis, Brian. "Lattice Simplices: Sufficiently Complicated." UKnowledge, 2019. https://uknowledge.uky.edu/math_etds/60.

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Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the fundamental parallelepiped associated to the simplex. We conclude with a proof-of-concept for using machine learning techniques in algebraic combinatorics. Specifically, we attempt to model the integer decomposition property of a family of lattice simplices using a neural network.
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Gay, Joël. "Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS209/document.

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La combinatoire algébrique est le champ de recherche qui utilise des méthodes combinatoires et des algorithmes pour étudier les problèmes algébriques, et applique ensuite des outils algébriques à ces problèmes combinatoires. L’un des thèmes centraux de la combinatoire algébrique est l’étude des permutations car elles peuvent être interprétées de bien des manières (en tant que bijections, matrices de permutations, mais aussi mots sur des entiers, ordre totaux sur des entiers, sommets du permutaèdre…). Cette riche diversité de perspectives conduit alors aux généralisations suivantes du groupe symétrique. Sur le plan géométrique, le groupe symétrique engendré par les transpositions élémentaires est l’exemple canonique des groupes de réflexions finis, également appelés groupes de Coxeter. Sur le plan monoïdal, ces même transpositions élémentaires deviennent les opérateurs du tri par bulles et engendrent le monoïde de 0-Hecke, dont l’algèbre est la spécialisation à q=0 de la q-déformation du groupe symétrique introduite par Iwahori. Cette thèse se consacre à deux autres généralisations des permutations. Dans la première partie de cette thèse, nous nous concentrons sur les matrices de permutations partielles, en d’autres termes les placements de tours ne s’attaquant pas deux à deux sur un échiquier carré. Ces placements de tours engendrent le monoïde de placements de tours, une généralisation du groupe symétrique. Dans cette thèse nous introduisons et étudions le 0-monoïde de placements de tours comme une généralisation du monoïde de 0-Hecke. Son algèbre est la dégénérescence à q=0 de la q-déformation du monoïde de placements de tours introduite par Solomon. On étudie par la suite les propriétés monoïdales fondamentales du 0-monoïde de placements de tours (ordres de Green, propriété de treillis du R-ordre, J-trivialité) ce qui nous permet de décrire sa théorie des représentations (modules simples et projectifs, projectivité sur le monoïde de 0-Hecke, restriction et induction le long d’une fonction d’inclusion).Les monoïdes de placements de tours sont en fait l’instance en type A de la famille des monoïdes de Renner, définis comme les complétés des groupes de Weyl (c’est-à-dire les groupes de Coxeter cristallographiques) pour la topologie de Zariski. Dès lors, dans la seconde partie de la thèse nous étendons nos résultats du type A afin de définir les monoïdes de 0-Renner en type B et D et d’en donner une présentation. Ceci nous conduit également à une présentation des monoïdes de Renner en type B et D, corrigeant ainsi une présentation erronée se trouvant dans la littérature depuis une dizaine d’années. Par la suite, nous étudions comme en type A les propriétés monoïdales de ces nouveaux monoïdes de 0-Renner de type B et D : ils restent J-triviaux, mais leur R-ordre n’est plus un treillis. Cela ne nous empêche pas d’étudier leur théorie des représentations, ainsi que la restriction des modules projectifs sur le monoïde de 0-Hecke qui leur est associé. Enfin, la dernière partie de la thèse traite de différentes généralisations des permutations. Dans une récente séries d’articles, Châtel, Pilaud et Pons revisitent la combinatoire algébrique des permutations (ordre faible, algèbre de Hopf de Malvenuto-Reutenauer) en terme de combinatoire sur les ordres partiels sur les entiers. Cette perspective englobe également la combinatoire des quotients de l’ordre faible tels les arbres binaires, les séquences binaires, et de façon plus générale les récents permutarbres de Pilaud et Pons. Nous généralisons alors l’ordre faibles aux éléments des groupes de Weyl. Ceci nous conduit à décrire un ordre sur les sommets des permutaèdres, associaèdres généralisés et cubes dans le même cadre unifié. Ces résultats se basent sur de subtiles propriétés des sommes de racines dans les groupes de Weyl qui s’avèrent ne pas fonctionner pour les groupes de Coxeter qui ne sont pas cristallographiques
Algebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems. One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron…). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the 0-Hecke monoid, whose algebra is the specialization at q=0 of Iwahori’s q-deformation of the symmetric group. This thesis deals with two further generalizations of permutations. In the first part of this thesis, we first focus on partial permutations matrices, that is placements of pairwise non attacking rooks on a n by n chessboard, simply called rooks. Rooks generate the rook monoid, a generalization of the symmetric group. In this thesis we introduce and study the 0-Rook monoid, a generalization of the 0-Hecke monoid. Its algebra is a proper degeneracy at q = 0 of the q-deformed rook monoid of Solomon. We study fundamental monoidal properties of the 0-rook monoid (Green orders, lattice property of the R-order, J-triviality) which allow us to describe its representation theory (simple and projective modules, projectivity on the 0-Hecke monoid, restriction and induction along an inclusion map).Rook monoids are actually type A instances of the family of Renner monoids, which are completions of the Weyl groups (crystallographic Coxeter groups) for Zariski’s topology. In the second part of this thesis we extend our type A results to define and give a presentation of 0-Renner monoids in type B and D. This also leads to a presentation of the Renner monoids of type B and D, correcting a misleading presentation that appeared earlier in the litterature. As in type A we study the monoidal properties of the 0-Renner monoids of type B and D : they are still J-trivial but their R-order are not lattices anymore. We study nonetheless their representation theory and the restriction of projective modules over the corresponding 0-Hecke monoids. The third part of this thesis deals with different generalizations of permutations. In a recent series of papers, Châtel, Pilaud and Pons revisit the algebraic combinatorics of permutations (weak order, Malvenuto-Reutenauer Hopf algebra) in terms of the combinatorics of integer posets. This perspective encompasses as well the combinatorics of quotients of the weak order such as binary trees, binary sequences, and more generally the recent permutrees of Pilaud and Pons. We generalize the weak order on the elements of the Weyl groups. This enables us to describe the order on vertices of the permutahedra, generalized associahedra and cubes in the same unified context. These results are based on subtle properties of sums of roots in Weyl groups, and actually fail for non-crystallographic Coxeter groups
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Books on the topic "Lattices and Combinatorics"

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Gerhard, Gierz, ed. Continuous lattices and domains. Cambridge, U.K: Cambridge University Press, 2003.

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Andrews, George E., Christian Krattenthaler, and Alan Krinik, eds. Lattice Path Combinatorics and Applications. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11102-1.

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Conway, John Horton. Sphere packings, lattices, and groups. 3rd ed. New York: Springer, 1999.

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Conway, John Horton. Sphere packings, lattices, and groups. New York: Springer-Verlag, 1988.

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A, Sloane N. J., ed. Sphere Packings, Lattices and Groups. New York, NY: Springer New York, 1988.

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Conway, John Horton. Sphere packings, lattices, and groups. 2nd ed. New York: Springer-Verlag, 1993.

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1951-, Terao Hiroaki, ed. Arrangements of hyperplanes. Berlin: Springer-Verlag, 1992.

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1951-, Terao Hiroaki, ed. Arrangements and hypergeometric integrals. Tokyo: Mathematical Society of Japan, 2001.

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Martinet, Jacques. Les Réseaux parfaits des espaces euclidiens. Paris: Masson, 1996.

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Orlik, Peter. Introduction to arrangements. Providence, R.I: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 1989.

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Book chapters on the topic "Lattices and Combinatorics"

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Deza, Michel Marie, and Monique Laurent. "Preliminaries on Lattices." In Algorithms and Combinatorics, 175–92. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-04295-9_13.

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Kung, Joseph P. S. "Combinatorics in finite lattices." In Lattice Theory: Special Topics and Applications, 195–229. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-06413-0_6.

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Kříž, Igor. "On Order-Perfect Lattices." In Algorithms and Combinatorics, 409–19. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-60406-5_35.

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Crapo, Henry H. "Möbius Inversion in Lattices." In Classic Papers in Combinatorics, 403–15. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4842-8_29.

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Geissinger, Ladnor. "Valuations on Distributive Lattices I." In Classic Papers in Combinatorics, 462–71. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4842-8_37.

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Geissinger, Ladnor. "Valuations on Distributive Lattices II." In Classic Papers in Combinatorics, 473–81. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4842-8_38.

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Geissinger, Ladnor. "Valuations on Distributive Lattices III." In Classic Papers in Combinatorics, 483–89. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4842-8_39.

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Berman, Joel. "On the Combinatorics of Free Algebras." In Lattices, Semigroups, and Universal Algebra, 13–19. Boston, MA: Springer US, 1990. http://dx.doi.org/10.1007/978-1-4899-2608-1_2.

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Deza, Michel Marie, and Monique Laurent. "Cut Lattices, Quasi h-Distances and Hilbert Bases." In Algorithms and Combinatorics, 381–93. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-04295-9_25.

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Deza, Michel Marie, and Monique Laurent. "L 1-Metrics from Lattices, Semigroups and Normed Spaces." In Algorithms and Combinatorics, 105–11. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-04295-9_8.

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Conference papers on the topic "Lattices and Combinatorics"

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Mishna, Marni. "Algorithmic Approaches for Lattice Path Combinatorics." In ISSAC '17: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2017. http://dx.doi.org/10.1145/3087604.3087664.

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Banderier, Cyril, and Michael Wallner. "Lattice paths of slope 2/5." In 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2014. http://dx.doi.org/10.1137/1.9781611973761.10.

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Unger, Wolfgang. "The Combinatorics of Lattice QCD at Strong Coupling." In The 32nd International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2015. http://dx.doi.org/10.22323/1.214.0192.

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Bhakta, Prateek, and Dana Randall. "Sampling Weighted Perfect Matchings on the Square-Octagon Lattice." In 2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2015. http://dx.doi.org/10.1137/1.9781611974324.5.

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Bernasconi, Anna, Antonio Boffa, Fabrizio Luccio, and Linda Pagli. "Two Combinatorial Problems on the Layout of Switching Lattices." In 2018 IFIP/IEEE International Conference on Very Large Scale Integration (VLSI-SoC). IEEE, 2018. http://dx.doi.org/10.1109/vlsi-soc.2018.8644855.

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Kasparian, Azniv. "Riemann-Roch Theorem and Mac Williams identities for an additive code with respect to a saturated lattice." In 2020 Algebraic and Combinatorial Coding Theory (ACCT). IEEE, 2020. http://dx.doi.org/10.1109/acct51235.2020.9383243.

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Buermann, Jan, and Jie Zhang. "Multi-Robot Adversarial Patrolling Strategies via Lattice Paths." In Twenty-Ninth International Joint Conference on Artificial Intelligence and Seventeenth Pacific Rim International Conference on Artificial Intelligence {IJCAI-PRICAI-20}. California: International Joint Conferences on Artificial Intelligence Organization, 2020. http://dx.doi.org/10.24963/ijcai.2020/582.

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Abstract:
In full-knowledge multi-robot adversarial patrolling, a group of robots have to detect an adversary who knows the robots' strategy. The adversary can easily take advantage of any deterministic patrolling strategy, which necessitates the employment of a randomised strategy. While the Markov decision process has been the dominant methodology in computing the penetration detection probabilities, we apply enumerative combinatorics to characterise the penetration detection probabilities. It allows us to provide the closed formulae of these probabilities and facilitates characterising optimal random defence strategies. Comparing to iteratively updating the Markov transition matrices, our methods significantly reduces the time and space complexity of solving the problem. We use this method to tackle four penetration configurations.
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LEE, EDMOND W. H., and M. V. VOLKOV. "ON THE STRUCTURE OF THE LATTICE OF COMBINATORIAL REES–SUSHKEVICH VARIETIES." In Proceedings of the International Conference. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812708700_0012.

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Pappi, Koralia N., Nestor D. Chatzidiamantis, and George K. Karagiannidis. "A combinatorial geometrical approach to the error performance of multidimensional finite lattice constellations." In 2012 IEEE Wireless Communications and Networking Conference (WCNC). IEEE, 2012. http://dx.doi.org/10.1109/wcnc.2012.6214435.

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Callanan, Jesse, Oladapo Ogunbodede, Maulikkumar Dhameliya, Jun Wang, and Rahul Rai. "Hierarchical Combinatorial Design and Optimization of Quasi-Periodic Metamaterial Structures." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-85914.

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As advanced manufacturing techniques such as additive manufacturing become widely available, it is of interest to investigate the potential advantages that arise when designing periodic metamaterials to achieve a specific desired behavior or physical property. Designing the fine scale detailed geometry of periodic metamaterials to achieve a specified behavior falls under the category of notoriously intractable inverse problems. To simplify solving the inverse problem, most relevant works represent metamaterials as periodic single unit cell structures repeated in regular lattices. Such representation simplifies modeling and simulation task but at the cost of possibly limiting the range of physical behaviors that can be achieved through the use of more than one unit cell structures. This article outlines a quasi-periodic representation that utilizes more than a single unit cell to generate periodic metamaterials. Additionally, a hierarchical optimization scheme to optimize the generating function for a quasi-periodic structure using the genetic algorithm (GA) and a barrier function interior point method is also sketched to solve the inverse problem. To demonstrate the utility of the proposed hierarchical optimization framework to solve quasi-periodic metamaterial inverse problem, a problem in which the objective is to minimize the total strain in the structure while subjected to weight and the total-size constraint is considered. We detail the overall computational approach in which geometric representation, optimization algorithms, and finite element analysis are coupled and report preliminary numerical experiments.
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