dc.contributor.author | Mortimer, Paul | |
dc.date.accessioned | 2016-06-21T10:55:23Z | |
dc.date.available | 2016-06-21T10:55:23Z | |
dc.date.issued | 2016-01-20 | |
dc.date.submitted | 2016-06-14T15:06:15.974Z | |
dc.identifier.citation | Mortimer, P. 2016: Lattice path enumeration on restricted domains, Queen Mary University of London. | en_US |
dc.identifier.uri | http://qmro.qmul.ac.uk/xmlui/handle/123456789/12992 | |
dc.description | PhD | en_US |
dc.description.abstract | This thesis concerns the enumeration and structural properties of lattice paths.
The study of Dyck paths and their characteristics is a classical combinatorial
subject. In particular, it is well-known that many of their characteristics are
counted by the Narayana numbers. We begin by presenting an explicit bijection
between Dyck paths with two such characteristics, peaks and up-steps at odd
height, and extend this bijection to bilateral Dyck paths.
We then move on to an enumeration problem in which we utilise the Kernel
method, which is a cutting-edge tool in algebraic combinatorics. However, while it
has proven extremely useful for nding generating functions when used with one
or two catalytic variables, there have been few examples where a Kernel method
has been successfully used in a general multivariate setting. Here we provide one
such example.
We consider walks on a triangular domain that is a subset of the triangular
lattice. We then specialise this by dividing the lattice into two directed sublattices
with di erent weights. Our central result on this model is an explicit formula
for the generating function of walks starting at a xed point in this domain and
5
6
ending anywhere within the domain. We derive this via use of the algebraic Kernel
method with three catalytic variables.
Intriguingly, the specialisation of this formula to walks starting in a fixed corner
of the triangle shows that these are equinumerous to bicoloured Motzkin paths, and
bicoloured three-candidate Ballot paths, in a strip of unite height. We complete
this thesis by providing bijective proofs for small cases of this result. | |
dc.description.sponsorship | Queen Mary Postgraduate Research Fund
Queen Mary University of London | en_US |
dc.language.iso | en | en_US |
dc.publisher | Queen Mary University of London | en_US |
dc.subject | Mathematics | en_US |
dc.subject | Dyck paths | en_US |
dc.subject | algebraic combinatorics | en_US |
dc.title | Lattice path enumeration on restricted domains | en_US |
dc.type | Thesis | en_US |
dc.rights.holder | The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author | |