A generalization of the quadrangulation relation to constellations and hypermaps

Constellations and hypermaps generalize combinatorial maps, i.e. embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) stating an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by using a result of Littlewood on factorization of characters. A combinatorial proof of Littlewood's result is also given. Furthermore, we show that coefficients in our relation are all positive integers, hinting possibility of a combinatorial interpretation. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps in Chapuy (2009).Comment: 19 pages, extended abstract published in the proceedings of FPSAC 201

The combinatorics of the Jack parameter and the genus series for topological maps

Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically

The singular values of the GUE (less is more)

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large n limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter ± 1/2. Similarly, we write the absolute value of the determinant of the n x n GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.National Science Foundation (U.S.) (Grant DMS–1035400)National Science Foundation (U.S.) (Grant DMS–1016125

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds

Aspects of Quantum Field Theory in Enumerative Graph Theory

While a quantum field theorist has many uses for mathematics of all kinds, the relationship between quantum field theory and mathematics is far too fluid in the world of modern research to be described as the simple provision of mathematical tools to physicists, as Feynman often framed it. Problems large and small of a seemingly purely mathematical nature often arise directly from a physical setting. In this thesis we focus on two combinatorial problems with deep physical motivations. The first of these is the Quadrangulation Conjecture of Jackson and Visentin, which asks for a bijective proof of an identity relating numbers of maps to numbers of maps which are quadrangulations. We provide a set of auxiliary bijections culminating in a bijection between maps with marked spanning trees and chord diagrams with partitions of the chords into a non-crossing part and a ‘genus-g’ part, and a bijection between these partitioned chord diagrams and four-regular maps with marked Euler tours. The second problem comes from the CHY integral formulation of tree-level Feynman integrals in supersymmetric Yang-Mills theory, but amounts to the enumeration of ways to decompose 4-regular graphs into pairs of edge-disjoint Hamiltonian cycles. We show that for any graph which is the edge-disjoint union of an arbitrary 2-regular graph and a cycle, there are at least (n−2)!/4 ways to decompose the result into two full cycles. Moreover, if the chosen 2-regular graph consists of only even cycles this bound improves to (n − 2)!/2. Further, if the graph consists only of 2-cycles, we obtain the exact number of decompositions, which is (1/2) (n−2)!!S_H^±(n/2−1,1), where S_H^±(a,b) is the so-called signed Hultman number. Interestingly, this combinatorial problem turns out to have further connections to the study of genomic rearrangements in bioinformatics