An involution on Dyck paths and its consequences

AbstractAn involution is introduced in the set of all Dyck paths of semilength n from which one re-obtains easily the equidistribution of the parameters ‘number of valleys’ and ‘number of doublerises’ and also the equidistribution of the parameters ‘height of the first peak’ and ‘number of returns’

Universality results for largest eigenvalues of some sample covariance matrix ensembles

For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small.Comment: 3 figures 47 pages Simulations have been included, a mistake in the computation of the variance has been corrected (Section 2.5

Refined Catalan and Narayana cyclic sieving

We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.Comment: Updated version, minor change

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

Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.Comment: LaTeX2e, 122 pages, includes 9 tikz figure