Abstract: The practical implementation of active discrete processes management in lattice structures requires preliminary structural and technical solutions as to how the device to be assembled and secured. The problems of constructing mathematical models for mechanical systems, in which joints are available or can be formed, are also relevant. In both cases, these tasks allow for a description with a help of a finite or countable graphs and, in particular, integer lattices.
This work is related to the development of special software systems analysis and touches upon a number of issues on the modeling and analysis of internal dependencies of discrete processes on the lattice structures.
The tasks are formulated in terms of combinatorial routes on square and triangular two-dimensional lattices. This makes it possible to successfully apply combinatorial methods of modeling.
Tools of generating functions and one of the most important classes of finite-difference equations - homogeneous recurrence relations with constant coefficients - are widely used in the study of the routes on the lattices and the related combinatorial numbers.
In the present paper, generating functions of a series of combinatorial numbers that satisfy the homogeneous recurrence relations with constant coefficients, are found.
The study presents geometric interpretations of the generating functions of some combinatorial numbers in terms of the routes on the lattices.
A clear link of generating functions of a two-dimensional lattice and its sublattices of a certain kind is revealed. As an example, the process of constructing generating functions of Fibonacci numbers with even and odd indices is provided.
The results of the work can be used in the analysis of topology of the informational networks and transportation structures. |