Esta dissertação descreve um jogo de baralho com caráter pedagógico, Resto Zero, o qual apresenta forte ligação com probabilidade, divisibilidade, análise combinatória e operações aritméticas elementares. Especificamente calculamos a probabilidade de alguns eventos principais que ocorrem no desenvolvimento do jogo. Apresentamos também uma relação do uso do Resto Zero aos anos/séries em que pode ser trabalhado. / In this dissertation we present and develop a simple game based upon a deck of cards which we call Residue Zero. We study and describe some characteristics of this game by observing its strong connections with probability, combinatorics and basic arithmetic operations. In particular, we compute the probability of several events that occur during the development of this game. We finally provide a relation of the scholar grades in which some features of this game could be worked out.

Esta dissertação descreve um jogo de baralho com caráter pedagógico, Resto Zero, o qual apresenta forte ligação com probabilidade, divisibilidade, análise combinatória e operações aritméticas elementares. Especificamente calculamos a probabilidade de alguns eventos principais que ocorrem no desenvolvimento do jogo. Apresentamos também uma relação do uso do Resto Zero aos anos/séries em que pode ser trabalhado. / In this dissertation we present and develop a simple game based upon a deck of cards which we call Residue Zero. We study and describe some characteristics of this game by observing its strong connections with probability, combinatorics and basic arithmetic operations. In particular, we compute the probability of several events that occur during the development of this game. We finally provide a relation of the scholar grades in which some features of this game could be worked out.

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT : In this thesis, a number of combinatorial objects are enumerated. Du and Yin as well as Shin and Zeng (by a different approach) proved an elegant formula for the number of labelled trees with respect to a given in degree sequence, where each edge is oriented from a vertex of lower label towards a vertex of higher label. We refine their result to also take the number of sources (vertices of in degree 0) or sinks (vertices of out degree 0) into account. We find formulas for the mean and variance of the number of sinks or sources in these trees. We also obtain a differential equation and a functional equation satisfied by the generating function for these trees. Analogous results for labelled trees with two marked vertices, related to functional digraphs, are also established. We extend the work to count reachable vertices, sinks and leaf sinks in these trees. Among other results, we obtain a counting formula for the number of labelled trees on n vertices in which exactly k vertices are reachable from a given vertex v and also the average number of vertices that are reachable from a specified vertex in labelled trees of order n. In this dissertation, we also enumerate certain families of set partitions and related tree-like structures. We provide a proof for a formula that counts connected cycle-free families of k set partitions of {1, . . . , n} satisfying a certain coherence condition and then establish a bijection between these families and the set of labelled free k-ary cacti with a given vertex-degree distribution. We then show that the formula also counts coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We extend the work to count coloured oriented cacti and coloured cacti. Noncrossing trees and related tree-like structures are also considered in this thesis. Specifically, we establish formulas for locally oriented noncrossing trees with a given number of sources and sinks, and also with given indegree and outdegree sequences. The work is extended to obtain the average number of reachable vertices in these trees. We then generalise the concept of noncrossing trees to find formulas for the number of noncrossing Husimi graphs, cacti and oriented cacti. The study is further extended to find formulas for the number of bicoloured noncrossing Husimi graphs and the number of noncrossing connected cycle-free pairs of set partitions. / AFRIKAANSE OPSOMMING : In hierdie tesis word ’n aantal kombinatoriese objekte geenumereer. Du en Yin asook Shin en Zeng (deur middel van ’n ander benadering) het ’n elegante formule vir die aantal geëtiketteerde bome met betrekking tot ’n gegewe ingangsgraadry, waar elke lyn van die nodus met die kleiner etiket na die nodus met die groter etiket toe georiënteer word. Ons verfyn hul resultaat deur ook die aantal bronne (nodusse met ingangsgraad 0) en putte (nodusse met uitgangsgraad 0) in ag te neem. Ons vind formules vir die gemiddelde en variansie van die aantal putte of bronne in hierdie bome. Ons bepaal verder ’n differensiaalvergelyking en ’n funksionaalvergelyking wat deur die voortbringende funksie van hierdie bome bevredig word. Analoë resultate vir geëtiketteerde bome met twee gemerkte nodusse (wat verwant is aan funksionele digrafieke), is ook gevind. Ons gaan verder voort deur ook bereikbare nodusse, bronne en putte in hierdie bome at te tel. Onder andere verkry ons ’n formule vir die aantal geëtiketteerde bome met n nodusse waarin presies k nodusse vanaf ’n gegewe nodus v bereikbaar is asook die gemiddelde aantal nodusse wat bereikbaar is vanaf ’n gegewe nodus. Ons enumereer in hierdie tesis verder sekere families van versamelingsverdelings en soortgelyke boom-vormige strukture. Ons gee ’n bewys vir ’n formule wat die aantal van samehangende siklus-vrye families van k versamelingsverdelings op {1, . . . , n} wat ’n sekere koherensie-vereiste bevredig, en ons beskryf ’n bijeksie tussen hierdie familie en die versameling van geëtiketteerde vrye k-êre kaktusse met ’n gegewe nodus-graad-verdeling. Ons toon ook dat hierdie formule ook gekleurde Husimi-grafieke tel waar blokke van dieselfde kleur nie insident met mekaar mag wees nie. Ons tel verder ook gekleurde georiënteerde kaktusse en gekleurde kaktusse. Nie-kruisende bome en soortgelyke boom-vormige strukture word in hierdie tesis ook beskou. On bepaal spesifiek formules vir lokaal georiënteerde nie-kruisende bome wat ’n gegewe aantal bronne en putte het asook nie-kruisende bome met gegewe ingangs- en uitgangsgraadrye. Ons gaan voort deur die gemiddelde aantal bereikbare nodusse in hierdie bome te bepaal. Ons veralgemeen dan die konsep van nie-kruisende bome en vind formules vir die aantal nie-kruisende Husimi-grafieke, kaktusse en georiënteerde kaktusse. Laastens vind ons ’n formule vir die aantaal tweegekleurde nie-kruisende Husimi-grafieke en die aantal nie-kruisende samehangende siklus-vrye pare van versamelingsverdelings.

This thesis concerns the use of labelled infinite trees to solve families of nested recursions of the form $R(n)=\sum_{i=1}^kR(n-a_i-\sum_{j=1}^{p_i}R(n-b_{ij}))+w$, where $a_i$ is a nonnegative integer, $w$ is any integer, and $b_{ij},k,$ and $p_i$ are natural numbers. We show that the solutions to many families of such nested recursions have an intriguing combinatorial interpretation, namely, they count nodes on the bottom level of labelled infinite trees that correspond to the recursion. Furthermore, we show how the parameters defining these recursion families relate in a natural way to specific structural properties of the corresponding tree families. We introduce a general tree ``pruning" methodology that we use to establish all the required tree-sequence correspondences.

This thesis concerns the use of labelled infinite trees to solve families of nested recursions of the form $R(n)=\sum_{i=1}^kR(n-a_i-\sum_{j=1}^{p_i}R(n-b_{ij}))+w$, where $a_i$ is a nonnegative integer, $w$ is any integer, and $b_{ij},k,$ and $p_i$ are natural numbers. We show that the solutions to many families of such nested recursions have an intriguing combinatorial interpretation, namely, they count nodes on the bottom level of labelled infinite trees that correspond to the recursion. Furthermore, we show how the parameters defining these recursion families relate in a natural way to specific structural properties of the corresponding tree families. We introduce a general tree ``pruning" methodology that we use to establish all the required tree-sequence correspondences.

Le but de cette thèse est d'établir des résultats énumératifs sur certaines classes de chemins et d'animaux. Ces résultats sont obtenus en appliquant la théorie des empilements de pièces développée par Viennot. Nous étudions les excursions discrètes (ou chemins de Dyck généralisés) de hauteur bornée; nous obtenons des résultats énumératifs qui interprètent combinatoirement et étendent des résultats de Banderier, Flajolet et Bousquet-Mélou. Nous décrivons et énumérons plusieurs classes de chemins auto-évitants, dits chemins faiblement dirigés. Ces chemins sont plus nombreux que les chemins prudents qui forment la classe naturelle la plus grande jusqu'alors. Nous calculons le périmètre de site moyen des animaux dirigés, prouvant des conjectures de Conway et Le Borgne. Enfin, nous obtenons des résultats nouveaux sur l'énumération des animaux de Klarner et les animaux multi-dirigés de Bousquet-Mélou et Rechnitzer. / The goal of this thesis is to prove enumerative results on some classes of lattice walks and animals. These results are applications of the theory of heaps of pieces developed by Viennot. We study discrete excursions (or generalized Dyck paths) with bounded height; we obtain enumerative results that give a combinatorial interpretation and extend results by Banderier, Flajolet and Bousquet-Mélou. We describe and enumerate several classes of self-avoiding walks called weakly directed walks. These classes are larger than the class of prudent walks, the largest natural class enumerated so far. We compute the average site perimeter of directed animals, proving conjectures by Conway and Le Borgne. Finally, we obtain new results on the enumeration of Klarner animals and multi-directed animals defined by Bousquet-Mélou and Rechnitzer.

The study of patterns in combinatorial structures has grown up in the past few decades to one of the most active trends of research in combinatorics. Historically, the study of permutations which are constrained by not containing subsequences ordered in various prescribed ways has been motivated by the problem of sorting permutations with certain devices. However, the richness of this notion became especially evident from its plentiful appearances in several very different disciplines, such as pure mathematics, mathematical physics, computer science,biology, and many others. In the last decades, similar notions of patterns have been considered on discrete structures other than permutations, such as integer sequences, lattice paths, graphs, matchings and set partitions. In the first part of this talk I will introduce the general framework of pattern posets and some classical problems about patterns. In the second part of this talk I will present some enumerative results obtained in my PhD thesis about patterns in permutations, lattice paths and matchings. In particular I will describe a generating tree with a single label for permutations avoiding the vincular pattern 1 - 32 - 4, a finite automata approach to enumerate lattice excursions avoiding a single pattern and some results about matchings avoiding juxtapositions and liftings of patterns.

The classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.

Cette thèse porte sur le polynôme de Tutte, étudié selon différents points de vue. Dans une première partie, nous nous intéressons à l’énumération des cartes planaires munies d’une forêt couvrante, ici appelées cartes forestières, avec un poids z par face et un poids u par composante non racine de la forêt. De manière équivalente, nous comptons selon le nombre de faces les cartes planaires C pondérées par TC(u + 1; 1), où TC désigne le polynôme de Tutte de C. Nous commençons par une caractérisation purement combinatoire de la série génératrice correspondante, notée F(z; u). Nous en déduisons que F(z; u) est différentiellement algébrique en z, c’est-à-dire que F satisfait une équation différentielle polynomiale selon z. Enfin, pour u ≥ -1, nous étudions le comportement asymptotique du n-ième coefficient de F(z; u). Nous observons une transition de phase en 0, avec notamment un régime très atypique en n-3 ln-2(n) pour u ϵ [-1; 0[, témoignant d’une nouvelle classe d’universalité pour les cartes planaires. Dans une seconde partie, nous proposons un cadre unificateur pour les différentes notions d’activités utilisées dans la littérature pour décrire le polynôme de Tutte.La nouvelle notion d’activité ainsi définie est appelée Δ-activité. Elle regroupe toutes les notions d’activité déjà connues et présente de belles propriétés, comme celle de Crapo qui définit une partition (adaptée à l’activité) du treillis des sous-graphes couvrants en intervalles. Nous conjecturons en dernier lieu que toute activité qui décrit le polynôme de Tutte et qui satisfait la propriété susmentionnée de Crapo peut être définie en termes de Δ-activités. / This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight z per face and a weight u per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps C weighted by TC(u + 1; 1), where TC is the Tutte polynomial of C.We begin by a purely combinatorial characterization of the corresponding generating function, denoted by F(z; u). We deduce from this that F(z; u) is differentially algebraic in z, that is, satisfies a polynomial differential equation in z. Finally, for u ≥ -1, we study the asymptotic behaviour of the nth coefficient of F(z; u).We observe a phase transition at 0, with a very unusual regime in n-3 ln-2(n) for u ϵ [-1; 0[, which testifiesa new universality class for planar maps. In the second part, we propose a framework unifying the notions of activity used in the literature to describe the Tutte polynomial. The new notion of activity thereby defined is called Δ-activity. It gathers all the notions of activities that were already known and has nice properties, as Crapo’s property that defines a partition of the lattice of the spanning subgraphs into intervals with respect to the activity. Lastly we conjecture that every activity that describes the Tutte polynomial and that satisfies Crapo’s property can be defined in terms of Δ-activity.

Cette thèse se divise en deux grandes parties, la première traite la combinatoire associée à l’ordre normal des opérateurs non-commutatifs et la seconde aborde des distributions symétriques du nombre de croisements et du nombre d’emboîtements, respectivement k-croisements et k-emboîtements, dans des structures combinatoires (partitions, permutations, permutations colorées, …). La première partie étudie l’ordre normal des opérateurs en termes de placements de tours. Nous étudions la forme de l’ordre normal en connectant deux opérateurs non-commutatifs D et U, et des polynômes orthogonaux spéciaux, et établissons des bijonctions entre les coefficients de (D+U)n et le nombre de placements de tours sur un diagramme de Ferrers. Nous donnons également des preuves combinatoires à des conjectures quantiques posées par des physiciens. Dans la seconde partie, nous définissons des statistiques, comme emboîtements et k-emboîtements, sur l’ensemble des permutations du groupe de Coxeter de type B. Nous donnons également des extensions au type B des résultats sur les croisements et les emboîtements, respectivement k-croisements et k-emboîtements dans les permutations de type A, en termes de distributions symétriques. De plus, nous étudions le lien entre les opérateurs non-commutatifs et ces statistiques. D’autres extensions de la distribution de ces statistiques sur les ensembles de partitions colorées et de permutations colorées de types A et B sont ainsi établies / This thesis is divided into two parts, the first deals with the combinatorics associated to the normal ordering form of noncommutative operators and the second addresses the symmetric distributions of the crossing numbers and nesting numbers, respectively k-crossings and k-nestings, in combinatorial structures (partitions, permutations, colored permutations, …). The first part studies the normal order of operators in terms of rook placements. We study the normal ordering form connecting two noncommutative operators D and U, and some special orthogonal polynomials, and establish bijonctions between coefficients of (D+U)n and rook placements in Ferrers diagrams. We also give combinatorial proofs and alternatives to some quantum conjectures posed by physicists. In the second part, we define the notions of statistics, nestings and k-nestings, on the sets of permutations of the Coxeter group of type B. We also give extensions to type B of the results of the crossings and nestings, respectivelu k-crossings and K-nestings in the set of permutations of type A, in terms of symmetric distributions. Likewise, we study the link between non-commutative operators and these statistics. Other extensions of the distribution of these statistics on the sets of colored partitions and colored permutations of type A and B are established