9 research outputs found
Asymptotics for incidence matrix classes
We define {\em incidence matrices} to be zero-one matrices with no zero rows
or columns. A classification of incidence matrices is considered for which
conditions of symmetry by transposition, having no repeated rows/columns, or
identification by permutation of rows/columns are imposed. We find asymptotics
and relationships for the number of matrices with ones in these classes as
.Comment: updated and slightly expanded versio
Asymptotic enumeration of incidence matrices
We discuss the problem of counting {\em incidence matrices}, i.e. zero-one
matrices with no zero rows or columns. Using different approaches we give three
different proofs for the leading asymptotics for the number of matrices with
ones as . We also give refined results for the asymptotic
number of incidence matrices with ones.Comment: jpconf style files. Presented at the conference "Counting Complexity:
An international workshop on statistical mechanics and combinatorics." In
celebration of Prof. Tony Guttmann's 60th birthda
Asymptotic enumeration of 2-covers and line graphs
In this paper we find asymptotic enumerations for the number of line graphs
on -labelled vertices and for different types of related combinatorial
objects called 2-covers.
We find that the number of 2-covers, , and proper 2-covers, , on
both have asymptotic growth where is the th Bell number, while the number of
restricted 2-covers, , restricted, proper 2-covers on , , and
line graphs , all have growth
In our proofs we use probabilistic arguments for the unrestricted types of
2-covers and and generating function methods for the restricted types of
2-covers and line graphs
Counting Gauge Invariants: the Plethystic Program
We propose a programme for systematically counting the single and multi-trace
gauge invariant operators of a gauge theory. Key to this is the plethystic
function. We expound in detail the power of this plethystic programme for
world-volume quiver gauge theories of D-branes probing Calabi-Yau
singularities, an illustrative case to which the programme is not limited,
though in which a full intimate web of relations between the geometry and the
gauge theory manifests herself. We can also use generalisations of
Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic
exponential. In due course, we also touch upon fascinating connections to Young
Tableaux, Hilbert schemes and the MacMahon Conjecture.Comment: 51 pages, 2 figures; refs updated, typos correcte
Enumerating 0-simple semigroups
Computational semigroup theory involves the study and implementation of algorithms to compute with semigroups. Efficiency is of central concern and often follows from the insight of semigroup theoretic results. In turn, computational methods allow for analysis of semigroups which can provide intuition leading to theoretical breakthroughs. More efficient algorithms allow for more cases to be computed and increases the potential for insight. In this way, research into computational semigroup theory and abstract semigroup theory forms a feedback loop with each benefiting the other.
In this thesis the primary focus will be on counting isomorphism classes of finite 0-simple semigroups. These semigroups are in some sense the building blocks of finite semigroups due to their correspondence with the Greens -classes of a semigroup. The theory of Rees 0-matrix semigroups links these semigroups to matrices with entries from 0-groups. Special consideration will be given to the enumeration of certain sub-cases, most prominently the case of congruence free semigroups. The author has implemented these enumeration techniques and applied them to count isomorphism classes of 0-simple semigroups and congruence free semigroups by order. Included in this thesis are tables of the number of 0-simple semigroups of orders less than or equal to 130, up to isomorphism. Also included are tables of the numbers of congruence free semigroups, up to isomorphism, with m Greenβs β-classes and n Greenβs β-classes for all mn less than or equal to 100, as well as for various other values of m,n. Furthermore a database of finite 0-simple semigroups has been created and we detail how this was done. The implementation of these enumeration methods and the database are publicly available as GAP code. In order to achieve these results pertaining to finite 0-simple semigroups we invoke the theory of group actions and prove novel combinatorial results. Most notably, we have deduced formulae for enumerating the number of binary matrices with distinct rows and columns up to row and column permutations.
There are also two sections dedicated to covers of E-unitary inverse semigroups, and presentations of factorisable orthodox monoids, respectively. In the first, we explore the concept of a minimal E-unitary inverse cover, up to isomorphism, by defining various sensible orderings. We provide examples of Clifford semigroups showing that, in general, these orderings do not have a unique minimal element. Finally, we pose conjectures about the existence of unique minimal E-unitary inverse covers of Clifford semigroups, when considered up to an equivalence weaker than isomorphism. In the latter section, we generalise the theory of presentations of factorisable inverse monoids to the more general setting of factorisable orthodox monoids. These topics were explored early in the authors doctoral studies but ultimately in less depth than the research on 0-simple semigroups
Asymptotics for incidence matrix classes
We define incidence matrices to be zero-one matrices with no zero rows or columns. We are interested in counting incidence matrices with a given number of ones, irrespective of the number of rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with n ones in some of these classes as n β β.</p
Asymptotics for incidence matrix classes
We define incidence matrices to be zero-one matrices with no zero rows or columns. We are interested in counting incidence matrices with a given number of ones, irrespective of the number of rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with n ones in some of these classes as n β β.