Browsing by Author "Selkirk, Sarah Jane"

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    On a generalisation of k-Dyck paths
    (Stellenbosch : Stellenbosch University, 2019-12) Selkirk, Sarah Jane; Wagner, Stephan; Prodinger, Helmut; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: (Refer to full text abstract for symbols that did not transfer correctly). We consider a family of non-negative lattice paths consisting of the step set f(1; 1); (1;􀀀k)g called k-Dyck paths, which are enumerated by the generalised Catalan numbers 1 (k+1)n+1 􀀀(k+1)n+1 n . By removing the non-negativity condition but restricting the path to stay above the line y = 􀀀t we obtain a family of lattice paths called kt-Dyck paths which are enumerated by `generalised generalised Catalan numbers t + 1 (k + 1)n + t + 1 (k + 1)n + t + 1 n : We provide proofs of the enumeration of these paths by means of a bijection, the kernel method, the cycle lemma, and the symbolic method. Analysis of parameters associated with the paths is also performed using symbolic equations { particularly the number of peaks, the number of valleys, and the number of returns. These kt-Dyck paths nd application in enumerating a family of walks in the quarter plane (Z 0 Z 0) with step set f(1; 1); (1;􀀀k +1); (􀀀k; 0)g. Such walks can be decomposed into ordered pairs of kt-Dyck paths and thus their enumeration can be proved via a simple bijection. Through this bijection some parameters in kt-Dyck paths are preserved. Finally, we discuss two different families of lattice paths, S-Motzkin and T- Motzkin paths, which are related to kt-Dyck paths when k = 2 along with t = 0 and t = 1. We provide bijections between these paths and other combinatorial objects, and perform analysis of parameters in these paths.