Combinatorial analysis of n-sized k-cycle substitutions with restricted cycle sizes

This study is devoted to combinatorial substitution schemes with various constraints on cycle sizes: lower bounds, upper bounds, and two-sided bounds.
For the proposed schemes, we solve several enumerative combinatorics problems: determining the number of possible outcomes, constructing direct numbered enumerations, solving numbering problems (establishing bijective correspondences between indices and types of outcomes), deriving probability distributions over the outcome sets, and developing a universal modeling procedure with specified probabilities.
All investigations are conducted using by the author’s enumerative method (EM), based on constructing a random process for the iterative formation and non-repetitive numbered enumeration of scheme outcomes. The outcomes of the first iteration—enumerating all valid cycle size compositions under the given constraints—are determined via schemes for placing indistinguishable particles into distinguishable cells under equivalent constraints. Subsequent iterations account for the distinctive features of our schemes’ interpretation within the placement framework, which involves distinguishable particles, indistinguishable cells, and consideration of particle order within each cell (starting from the particle with the minimum number).
In addition to the direct analysis of the schemes following the EM framework, we propose deriving some results by recalculating them from the outcomes of a similar analysis of more general, previously studied schemes with fewer restrictions on the relevant characteristics.