Combinatorial analysis of the scheme of dual repeated transfer

In the class of schemes for dividing particles into parts of given sizes, for the considered scheme with distinguishable particles and taking into account the order of the division parts (scheme $A$), a probabilistic model is constructed for a complete numbered listing of its outcomes, based on which it is studied in the following directions of enumerative combinatorics: finding their number, establishing an one-to-one correspondence between the numbers and types of its outcomes called the numbering problem in direct and inverse statements, finding probabilities on the set of its outcomes and proposing an algorithm for their modelling. Schemes of this class differ in the quality of their constituent elements (particles and dividing parts) in terms of their distinguishability. The scheme $A$ in this class has outcomes with the greatest differentiation, which makes it possible to obtain the outcomes of remaining schemes of this class by algorithmic procedures that lead to a certain groupings of its outcomes. To organize the possibility of recalculating from the results of the analysis of the scheme $A$ the corresponding results of other schemes of this class that requires separate consideration in each scheme, the model of the scheme $A$ is constructed with enumerations divided into stages, which separately take into account the distinguishability between the dividing parts and particles. The purpose of the article is to analyze the scheme $A$ in the form of obtaining analytical relations and constructing procedures and algorithms in the indicated directions of enumerative combinatorics and preparing its results and carrying out the corresponding recalculation for schemes of this class.