Bi-banded Paths, a Bijection and the Narayana Numbers

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the `TASEP' stochastic process.Comment: 10 pages, 5 figure

Lattice path enumeration on restricted domains

PhDThis 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.Queen Mary Postgraduate Research Fund Queen Mary University of Londo

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type. This framework also leads to random matrix models for boolean, monotone and s-free independences.Comment: 54 pages, the main change is that we describe the asymptotic joint distributions of symmetric blocks of a larger class of Hermitian random matrices (the former version covered only Gaussian random matrices

Bi-banded paths, a bijection and the Narayana numbers

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the 'TASEP' stochastic process