Generalized allocation scheme with cell occupancies from a fixed finite set

We consider a generalized scheme of allocation of n particles (elements) over unordered cells (components) under the condition that the number of particles in each cell belongs to a fixed finite set A of positive integers. A new asymptotic estimates for the total number In(A) of variants of allocations of n particles are obtained under some conditions on the set A; these estimates have an explicit form (up to equivalence). Some examples of combinatorial-probabilistic character are given to illustrate by particular cases the notions introduced and results obtained. For previously known theorems on the convergence to the normal law of the total number of components and numbers of components with given cardinalities the norming parameters are obtained in the explicit form without using roots of algebraic or transcendent equations.

Limit Poisson law for the distribution of the number of components in generalized allocation scheme

We consider problems on the convergence of distributions of the total number of components and numbers of components with given volume to the Poisson law. Sufficient conditions of such convergence are given. Our results generalize known statemets on the limit Poisson laws of the number of components (cycles, unrooted and rooted trees, blocks and other structures) in the corresponding generalized of allocation schemes.