Аннотация:

A discrete dynamical system called a binary chain of contours is considered. The system belongs to a class of dynamic systems developed by A.P. Buslaev with aim of creating traffic models with network structure and such that analytical results may be obtained for these systems.
The system under consideration contains a finite set of circumferences forming a closed chain.  There is a particle in any circumference. At any discrete moment, the particle is in a cell -  the lower cell or in the upper cell alternately. Delays in particle movement are due to the condition that two particle cannot pass simultaneously through the same node -  the common point of the circumferences. It was previously found that no later than at discrete moment of time following the initial moment the system results in the state of a cycle -  a closed trajectory in the system state space. The ratio of the number of particle movements during a cycle to the period -  duration of the cycle is called the average velocity of the particle. It was previously proved that, during a cycle, all particles move with the same average velocity and a formula was obtained for the spectrum of average velocities, i.e., the set of all possible values of the average velocities for  different initial states with a prescribed number of circumferences. In this paper, combinatorial formulas are obtained that allow, for any value belonging to the spectrum, to compute the total number of states belonging to cycles with this value of the average velocity and the number of non-recurrent states from that the system results in  states of cycles. Combinatorial problems for systems of the type under consideration have not been studied previously. The combinatorial formulas may be useful to plan simulation experiments or, for example, in considering theoretical problems related to the nature of the change in the entropy of a dynamic system.