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The scheme of allocating r distinguishable particles into n indistinguishable cells with k non-empty cells is studied along the directions of enumerative combinatorics. They are the direct enumeration of the outcomes of the scheme with a particular discipline of their numbering, finding their numbers, establishing a one-to-one correspondence of the types of outcomes with their numbers, also known as numbering problem in direct and inverse statements, and modeling of outcomes of the scheme.

COMBINATORIAL ANALYSIS IN THE SCHEME OF ALLOCATION OF DISTINGUISHABLE PARTICLES INTO INDISTINGUISHABLE CELLS WITH A GIVEN NUMBER OF NON-EMPTY CELLS