Generalized narayana number sequences, their polynomials, applications and pascal triangle

In this thesis, New Narayana polynomials are formed and the relationship between the coefficients of these polynomials and Pascal's triangle is investigated. Similarly, Gaussian Narayana numbers and polynomials are constructed. The coefficients obtained by taking the 1st and 2nd degree derivatives of these polynomials, respectively, are similarly related to Pascal's triangle. Then, by examining the study, which we aim to make more general, on the k-Narayana sequence, the self similarities in the Narayana sequence in addition to the cycles of the sequences in the Pascal 3-triangle are shown with flip graphs on the k-Narayana sequence. In order to simplify the characterization of free group structures underlying Pascal 3-triangle cycles, Betti numbers are introduced and some properties of Narayana and k-Narayana numbers and some theorems about these numbers are given. In addition, the Narayana triangle was created inspired by Hosoya's triangle and it was geometrically shown on the plane in order to embody the basic features of this triangle. Finally, the Narayana number sequence is studied on the module m, and based on the definition of the Narayana orbit, the length of the period of the Narayana orbit for the 2-generator groups is found. In addition, the study includes the length of the Narayana period on polyhedral and binary polyhedral groups, which are 2-generator groups, and the basic properties and applications by associating them.