Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

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Browsing Combinatorics and Optimization by Author "Goulden, Ian"

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    Combinatorial Methods for Enumerating Maps in Surfaces of Arbitrary Genus
    (University of Waterloo, 2016-06-10) Chan, Aaron C.S.; Goulden, Ian
    The problem of map enumeration is one that has been studied intensely for the past half century. Early work on this subject included the works of Tutte for various types of rooted planar maps and the works of Brown for non-planar maps. Furthermore, the works of Bender, Canfield, and Richmond as well as Bender and Gao give asymptotic results for the enumeration of various types of maps. This subject also attracted the attention of physicists when they independently discovered that map enumeration can be applied to quantum field theory. The results of 't Hooft established the connection between matrix integration and map enumeration, which allowed algebraic techniques to be used. Other examples of this application can be found in the papers of Itzykson and Zuber. One result of particular significance is the Harer-Zagier formula, which gives the genus series for maps with one vertex. This result has been proved many times in the literature, a selection of which includes the proofs of Goulden and Nica, Itzykson and Zuber, Jackson, Kerov, Kontsevich, Lass, Penner, and Zagier. An extension of this result to locally orientable maps on one vertex can be found in Goulden and Jackson, while another extension to two vertex maps can be found in Goulden and Slofstra. In this thesis, we will extend the combinatorial techniques used in the papers of Goulden and Nica and Goulden and Slofstra, so that they can be applied to maps with an arbitrary number of vertices, when the graph being embedded is a tree with loops and multiple edges. This involves defining a new set of combinatorial objects that extends the ones used in Goulden and Slofstra, and develop new techniques for handling these objects. Furthermore, we will simplify some of the techniques and results in the existing literature. Finally, we seek to relate the techniques used in this thesis to techniques in other map enumeration problems, and briefly discuss the potential of applying our techniques to those problems.
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    Computing the Residue Class of Partition Numbers
    (University of Waterloo, 2016-09-14) Shuldiner, Pavel; Goulden, Ian
    In 1919, Ramanujan initiated the study of congruence properties of the integer partition function $p(n)$ by showing that $$p(5n+4) \equiv 0 \mod{5}$$ and $$p(7n+5) \equiv 0 \mod{7}$$ hold for all integers $n$. These results attracted a lot of interest in the mathematical community and inspired other mathematicians to investigate the divisibility of various classes of integer partitions. The purpose of this thesis is to illustrate the use of generating series in the study of the residue classes of integer partition values. We begin by presenting the work of Mizuhara, Sellers and Swisher in 2015 on the residue classes of restricted plane partitions numbers. Next, we introduce Ramanujan's Conjecture regarding Ramanujan Congruences. Moreover, we use modular forms to present Ahlgren and Boylan's resolution of Ramanujan's Conjecture from 2003. Then, we discuss the open problems surrounding the distribution of the integer partitions values into residue classes and present Judge, Keith and Zanello's work from 2015 on the the distribution of the parity of the partition function. We continue by introducing $m-$ary partitions and provide an account of Andrews, Fraenkel and Sellers' work from 2015 and 2016 which yielded a complete characterization of the congruence classes of $m-$ary partitions with and without gaps. Finally, we present new results regarding the complete characterization of the residue classes of coloured $m-$ary partitions with and without gaps.
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    Induction Relations in the Symmetric Groups and Jucys-Murphy Elements
    (University of Waterloo, 2018-08-16) Chan, Kelvin Tian Yi; Goulden, Ian
    Transitive factorizations faithfully encode many interesting objects. The well-known ones include ramified coverings of the sphere and hypermaps. Enumeration of specific classes of such objects have been known for quite some time now. Hurwitz numbers, monotone Hurwitz numbers and hypermaps numbers were discovered using different techniques. Recently, Carrell and Goulden found a unified algebraic approach to count these objects in genus 0. Jucys-Murphy elements and centrality play important roles in establishing induction relations. Such a method is interesting in its own right. Its corresponding combinatorial decomposition is however intriguingly mysterious. Towards a understanding of direct combinatorial analysis of multiplication of arbitrary permutations, we consider methods, especially operators on symmetric functions, and related problems in symmetric groups.