This thesis examines the problem of reconstructing multiple discrete 2D objects, represented by a set of cells arranged in an m × n grid, from their projections. The objects being constructed are disjoint, hv-convex polyominoes, each of which has a separate colour. The main results presented here are two algorithms for unordered C-colour reconstruction that have time complexities of O(C^2n^{2C +1}m^{2C +1}) and O(C^2 min(n^{2C}, m^{2C})nm), an ordered C-colour reconstruction algorithm that is O(Cmin(n^{2C}, m^{2C})nm), and an NP-completeness proof when the number of colours is unbounded.

This thesis examines the problem of reconstructing multiple discrete 2D objects, represented by a set of cells arranged in an m × n grid, from their projections. The objects being constructed are disjoint, hv-convex polyominoes, each of which has a separate colour. The main results presented here are two algorithms for unordered C-colour reconstruction that have time complexities of O(C^2n^{2C +1}m^{2C +1}) and O(C^2 min(n^{2C}, m^{2C})nm), an ordered C-colour reconstruction algorithm that is O(Cmin(n^{2C}, m^{2C})nm), and an NP-completeness proof when the number of colours is unbounded.

We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, we propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s, A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations.

In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes.

Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.

Dans cette thèse nous étudions la caractérisation et l'énumération de polyominos définis par des contraintes de convexité et ou d'évitement de motifs. Nous nous intéressons à l'énumération des polyominos k-convexes selon le semi périmètre, qui n'était connue que pour k=1,2. Nous énumérons une sous classe, les polyominos k-parallélogrammes, grâce à une décomposition récursive dont nous déduisons la fonction génératrice qui est rationnelle. Cette fonction génératrice s'exprime à l'aide des polynômes de Fibonacci, ce qui nous permet d'en déduire une bijection avec les arbres planaires ayant une hauteur inférieure ou égale à k+2. Dans la deuxième partie, nous examinons la notion d'évitement de motif, qui a été essentiellement étudiée pour les permutations. Nous introduisons ce concept dans le contexte de matrices de permutations et de polyominos. Nous donnons des définitions analogues à celles données pour les permutations et nous explorons ses propriétés ainsi que celles du poste associé. Ces deux approches peuvent être utilisées pour traiter des problèmes ouverts sur les polyominos ou sur d'autres objets combinatoires. / In this thesis, we consider the problem of characterising and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well-known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polynomials, we describe a bijection between the class of k-parallelogram polyominoes and the class of planted planar trees having height less than k+3. In the second part of the thesis we examine the notion of pattern avoidance, which has been extensively studied for permutations. We introduce the concept of pattern avoidance in the context of matrices, more precisely permutation matrices and polyomino matrices. We present definitions analogous to those given for permutations and in particular we define polyomino classes, i.e. sets downward closed with respect to the containment relation. So, the study of the old and new properties of the redefined sets of objects has not only become interesting, but it has also suggested the study of the associated poset. In both approaches our results can be used to treat open problems related to polyominoes as well as other combinatorial objects.

Lam Wing Him Zephyr. / "Architecture Department, Chinese University of Hong Kong, Master of Architecture Programme 2004-2005, design report." / research / Chapter 1. --- Polyominoes / Chapter 1.1 --- What is polyominoes / Chapter 1.2 --- Shapes / Chapter 2. --- Games / Chapter 2.1 --- Areas / Chapter 2.2 --- Volumes / Chapter 3. --- Properties / Chapter 3.1 --- Axis / Chapter 3.2 --- Nodes / Chapter 3.3 --- Joints / Chapter 3.4 --- Rectangular frames / Chapter 3.5 --- Patternings / Chapter 4. --- Combinations / Chapter 4.1 --- Formation of blocks / Chapter 4.2 --- Blocks from 2D / Chapter 4.3 --- Blocks from 3D / Chapter 5. --- Cases / Chapter 5.1 --- Case study 01 / Chapter 5.2 --- Case study 02 / Chapter 5.3 --- Case study 03 / design / Chapter 6. --- Backgrounds / Chapter 6.1 --- Site / Chapter 6.2 --- Programs / Chapter 6.3 --- Selections / Chapter 6.4 --- Spatial variations / Chapter 7. --- Prefabrications / Chapter 7.1 --- Modules / Chapter 7.2 --- Modular formations / Chapter 7.3 --- Infills / Chapter 8. --- Applications / Chapter 8.1 --- Formations / Chapter 8.2 --- "Plans, sections & elevations" / Chapter 8.3 --- Views / Chapter 8.4 --- Key developments / Chapter 8.5 --- Models / Appendix

Cette thèse porte sur l’interprétation combinatoire des probabilitésde l’état stationnaire de l’ASEP par les tableaux escaliers, sur les arbresnon-ambigus et sur les polyominos parallélogrammes périodiques.Dans une première partie, nous étudions l’ansatz matriciel de Derrida,Evans, Hakim et Pasquier. Toute solution de ce système d’équation permet decalculer les probabilités stationnaires de l’ASEP. Nos travaux définissent denouvelles récurrences équivalentes à celles de l’ansatz matriciel. En définissantun algorithme d’insertion sur les tableaux escaliers, nous montrons combinatoirementet simplement qu’ils les satisfont. Nous faisons de même pour l’ASEPà deux particules. Enfin, nous énumérons les coins dans les tableaux associésà l’ASEP, nous permettant ainsi de donner le nombre moyen de transitionspossibles depuis un état de l’ASEP.Dans une deuxième partie, nous calculons de jolies formules pour les sériesgénératrices des arbres non-ambigus, desquelles nous déduisons des formulesd’énumérations. Puis, nous interprétons bijectivement certains de ces résultats.Enfin, nous généralisons les arbres non-ambigus à toutes les dimensions finies.Dans la dernière partie, nous construisons une structure arborescente surles polyominos parallélogrammes périodiques, inspirée des travaux de Boussicault,Rinaldi et Socci. Cela nous permet de calculer facilement leur sériegénératrice selon la hauteur et la largeur ainsi que deux nouvelles statistiques :la largeur intrinsèque et la hauteur de recollement intrinsèque. Enfin, nousétudions l’ultime périodicité de leur série génératrice selon l’aire. / This thesis deals with a combinatorial interpretation of the stationnarydistribution of the ASEP given by staircase tableaux and studiestwo combinatorial objects : non-ambiguous trees and periodic parallelogrampolyominoes.In the first part, we study the matrix ansatz introduced by Derrida, Evans,Hakim and Pasquier. Any solution of this equation system can be used tocompute the stationnary probabilities of the ASEP. Our work defines newrecurrences equivalent to the matrix ansatz. By defining an insertion algorithmfor staircase tableaux, we prove combinatorially and easily that they satisfyour new recurrences. We do the same for the ASEP with two types of particles.Finally, we enumerate the corners of the tableaux related to the ASEP, whichgives the average number of transitions from a state of the ASEP.In the second part, we compute nice formulas for the generating functionsof non-ambiguous trees, from which we deduce enumeration formulas. Then, wegive a combinatorial interpretation of some of our results. Lastly, we generalisenon-ambiguous trees to every finite dimension.In the last part, we define a tree structure in periodic parallelogram polyominoes,motivated by the work of Boussicault, Rinaldi and Socci. It allowsus to compute easily the generating function with respect to the height andthe width as well as two new statistics : the intrinsic width and the intrinsicgluing height. Finally, we investigate the ultimate periodicity of the generatingfunction with respect to the area.