The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}.

<p>Whether the growth series of a group is a rational function is investigated in this paper.Parry showed certain torus bundle groups of even trace exhibits rational growth, and thisresult has been extended by the author, Turbo Ho and Mark Pengitore. In this paper, bothresults are combined into a single proof used in [1], and the result is pushed further into thenegative case not covered in earlier works</p>

<p>As stressed by the Society for Industrial and Applied Mathematics (SIAM): Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Combinatorial optimization focuses on problems arising from discrete structures such as graphs and polyhedra. This thesis deals with extremal graphs and strings and focuses on two problems: the Erdos' problem on multiplicities of complete subgraphs and the maximum number of distinct squares in a string.<br />The first part of the thesis deals with strengthening the bounds for the minimum proportion of monochromatic t cliques and t cocliques for all 2-colourings of the edges of the complete graph on n vertices. Denote by k_t(G) the number of cliques of order t in a graph G. Let k_t(n) = min{k_t(G)+k_t(\overline{G})} where \overline{G} denotes the complement of G of order n. Let c_t(n) = {k_t(n)} / {\tbinom{n}{t}} and c_t be the limit of c_t(n) for n going to infinity. A 1962 conjecture of Erdos stating that c_t = 2^{1-\tbinom{t}{2}} was disproved by Thomason in 1989 for all t > 3. Tighter counterexamples have been constructed by Jagger, Stovicek and Thomason in 1996, by Thomason for t < 7 in 1997, and by Franek for t=6 in 2002. We present a computational framework to investigate tighter upper bounds for small t yielding the following improved upper bounds for t=6,7 and 8: c_6 \leq 0.7445 \times 2^{1- \tbinom{6}{2}}, c_7\leq 0.6869\times 2^{1- \tbinom{7}{2}}, and c_8 \leq 0.7002\times 2^{1- \tbinom{8}{2}}. The constructions are based on a large but highly regular variant of Cayley graphs for which the number of cliques and cocliques can be expressed in closed form. Considering the quantity e_t=2^{\tbinom{t}{2}-1} c_t, the new upper bound of 0.687 for e_7 is the first bound for any e_t smaller than the lower bound of 0.695 for e_4 due to Giraud in 1979.<br />The second part of the thesis deals with extremal periodicities in strings: we consider the problem of the maximum number of distinct squares in a string. The importance of considering as key variables both the length n and the size d of the alphabet is stressed. Let (d,n)-string denote a string of length n with exactly d distinct symbols. We investigate the function \sigma_d(n) = max {s(x) | x} where s(x) denotes the number of distinct primitively rooted squares in a (d,n)-string x. We discuss a computational framework for computing \sigma_d(n) based on the notion of density and exploiting the tightness of the available lower bound. The obtained computational results substantiate the hypothesized upper bound of n-d for \sigma_d(n). The structural similarities with the approach used for investigating the Hirsch bound for the diameter of a polytope of dimension d having n facets is underlined. For example, the role played by (d,2d)-polytope was presented in 1967 by Klee and Walkup who showed the equivalency between the Hirsch conjecture and the d-step conjecture.</p> / Doctor of Philosophy (PhD)

Thesis (MSc)--Stellenbosch University, 2015 / ENGLISH ABSTRACT : Two applications of analytic techniques to combinatorial problems with number-theoretic flavours are shown. The first is an application of the real saddle point method to derive second-order asymptotic expansions for the number of solutions to the signum equation of a general class of sequences. The second is an application of more elementary methods to yield asymptotic expansions for the number of partitions of a large integer into powers of an integer b where each part has bounded multiplicity. / AFRIKAANSE OPSOMMING : Ons toon twee toepassings van analitiese tegnieke op kombinatoriese probleme met getalteoretiese geure. Die eerste is ’n toepassing van die reële saalpuntmetode wat tweede-orde asimptotiese uitbreidings vir die aantal oplossings van die ‘signum’ vergelyking vir ’n algemene klas van rye aflewer. Die tweede is ’n toepassing van meer elementêre metodes wat asimptotiese uitbreidings vir die aantal partisies van ’n groot heelgetal in magte van ’n heelgetal b, waar elke deel ’n begrensde meervoudigheid het, aflewer

A map is an embedding of the vertices and edges of a graph into a compact 2-manifold such that the remainder of the surface has components homeomorphic to open disks. With the goal of proving the Four Colour Theorem, Tutte began the field of map enumeration in the 1960's. His methods included developing the edge deletion decomposition, developing and solving a recurrence and functional equation based on this decomposition, and developing the medial bijection between two equinumerous infinite families of maps. Beginning in the 1980's Jackson, Goulden and Visentin applied algebraic methods in enumeration of non-planar and non-orientable maps, to obtain results of interest for mathematical physics and algebraic geometry, and the Quadrangulation Conjecture and the Map-Jack Conjecture. A special case of the former is solved by Tutte's medial bijection. The latter uses Jack symmetric functions which are a topic of active research. In the 1960's Walsh and Lehman introduced a method of encoding orientable maps. We develop a similar method, based on depth first search and extended to non-orientable maps. With this, we develop a bijection that extends Tutte's medial bijection and partially solves the Quadrangulation Conjecture. Walsh extended Tutte's recurrence for planar maps to a recurrence for all orientable maps. We further extend the recurrence to include non-orientable maps, and express it as a partial differential equation satisfied by the generating series. By appropriately interpolating the differential equation and applying the depth first search method, we construct a parameter that empirically fulfils the conditions of the Map-Jack Conjecture, and we prove some of its predicted properties. Arques and Beraud recently obtained a continued fraction form of a specialisation of the generating series for maps. We apply the depth search method with an ordinary differential equation, to construct a bijection whose existence is implied by the continued fraction.

A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.

This thesis will study a variety of problems in graph theory. Initially, the focus will be on finding minimal degree conditions which guarantee the existence of various subgraphs. These subgraphs will all be formed of cycles, and this area of work will fall broadly into two main categories. First to be considered are cycles in edge-coloured graphs and, in particular, two questions of Li, Nikiforov and Schelp. It will be shown that a 2-edge-coloured graph with minimal degree at least 3n/4 either is isomorphic to the complete 4-partite graph with classes of order n/4, or contains monochromatic cycles of all lengths between 4 and n/2 (rounded up). This answers a conjecture of Li, Nikiforov and Schelp. Attention will then turn to the length of the longest monochromatic cycle in a 2-edge-coloured graph with minimal degree at least cn. In particular, a lower bound for this quantity will be proved which is asymptotically best possible. The next chapter of the thesis then shows that a hamiltonian graph with minimal degree at least (5-sqrt7)n/6 contains a 2-factor with two components. The thesis then concludes with a chapter about X_H, which is the number of copies of a graph H in the random graph G(n,p). In particular, it will be shown that, for a connected graph H, the value of X_H modulo k is approximately uniformly distributed, provided that k is not too large a function of n.

The simple relational structures form the units, or atoms, upon which all other relational structures are constructed by means of the substitution decomposition. This decomposition appears to have first been introduced in 1953 in a talk by Fraïssé, though it did not appear in an article until a paper by Gallai in 1967. It has subsequently been frequently rediscovered from a wide variety of perspectives, ranging from game theory to combinatorial optimization. Of all the relational structures - a set which also includes graphs, tournaments and posets - permutations are receiving ever increasing amounts of attention. A simple permutation is one that maps every nontrivial contiguous set of indices to a set of indices that is never contiguous. Simple permutations and intervals of permutations are important in biomathematics, while permutation classes - downsets under the pattern containment order - arise naturally in settings ranging from sorting to algebraic geometry. We begin by studying simple permutations themselves, though always aim to establish this theory within the broader context of relational structures. We first develop the technology of "pin sequences", and prove that every sufficiently long simple permutation must contain either a long horizontal or parallel alternation, or a long pin sequence. This gives rise to a simpler unavoidable substructures result, namely that every sufficiently long simple permutation contains a long alternation or oscillation. ErdÅ s, Fried, Hajnal and Milner showed in 1972 that every tournament could be extended to a simple tournament by adding at most two additional points. We prove analogous results for permutations, graphs, and posets, noting that in these three cases we may need to extend a structure by adding (n+1)/2 points in the case of permutations and posets, and logâ (n+1) points in the graph case. The importance of simple permutations in permutation classes has been well established in recent years. We extend this knowledge in a variety of ways, first by showing that, in a permutation class containing only finitely many simple permutations, every subset defined by properties belonging to a finite "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time. Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based.

In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for counting or sampling the elements of either set. The Eulerian orientations of a number of classes of planar lattices are of practical significance as they correspond to configurations of certain models studied in statistical physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a general Eulerian graph is #P-complete and demonstrated that the problem of sampling an Eulerian orientation can be reduced to the tractable problem of sampling a perfect matching of a bipartite graph. We present a proof that this problem remains #Pcomplete when the input is restricted to being a planar graph, and analyse a natural algorithm for generating random Eulerian orientations of one of the afore-mentioned planar lattices. Moreover, we make some progress towards classifying the range of planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class of planar graphs for which the algorithm will always take an exponential amount of time to converge. The problem of counting the Euler tours of undirected graphs has proven to be less amenable to analysis than that of Eulerian orientations. Although it has been known for many years that the number of Euler tours of any directed graph can be computed in polynomial time, until recently very little was known about the complexity of counting Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is #P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel graphs, there are no known tractable cases, nor are there any good reasons to believe the problem to be intractable. Moreover, despite several unsuccessful attempts, there has been no progress made on the question of approximability. Indeed, this problem was considered to be one of the more difficult open problems in approximate counting since long before the complexity of exact counting was resolved. By considering a randomised input model, we are able to show that a very simple algorithm can sample or approximately count the Euler tours of almost every d-in/d-out directed graph in expected polynomial time. Then, we present some partial results towards showing that this algorithm can be used to sample or approximately count the Euler tours of almost every 2d-regular graph in expected polynomial time. We also provide some empirical evidence to support the unproven conjecture required to obtain this result. As a sideresult of this work, we obtain an asymptotic characterisation of the distribution of the number of Eulerian orientations of a random 2d-regular graph.

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through the 11-cube, but do not discover any significant discernible patterns. We then dedicate the rest of our work to generalizing the notion of the Jacobian, specifically to be sensitive to edge directions. We conclude with a conjecture describing the form of the directed Jacobian of the directed $n$-topology. We hope for this work to be useful for theoretical particle physics and for graph theory in general.