Asymptotic expansions for the distribution of the number of components in random mappings and partitions
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A. N. Timashov
Abstract
We consider the class of all nn single-valued mappings of an n-element set into itself. Assuming that all such mappings have the same probabilities equal to n–n, we investigate the distribution of the random variable νn equal to the number of connected components in such random mapping. We obtain asymptotic estimates of the probability P{νn = N} as n, N → ∞ in such a way that the ratio N/ln n does not tend to 0 and infinity. In the case where 
as n → ∞, we obtain a complete asymptotic expansion of this probability.
A similar expansion is obtained for the probability P{ξn = N}, where ξn is the random variable equal to the number of cycles in a permutation randomly and equiprobably chosen from the set of all n! permutations of degree n, and also for the probability P{θn = N}, where θn is the number of blocks in a random partition of a set with n elements.
© de Gruyter 2011
Articles in the same Issue
- Prohibitions in discrete probabilistic statistical problems
- An asynchronous double stochastic flow with initiation of superfluous events
- Asymptotic expansions for the distribution of the number of components in random mappings and partitions
- Symmetric linear spaces of graphs
- On automorphisms of commutative Moufang loops
- On the mixing properties of operations in a finite field
- The predicate method to construct the Post lattice
- Average complexity of searching for identical objects in random nonuniform databases
- On construction of efficient algorithms for solving systems of polynomial Boolean equations by testing a part of variables
Articles in the same Issue
- Prohibitions in discrete probabilistic statistical problems
- An asynchronous double stochastic flow with initiation of superfluous events
- Asymptotic expansions for the distribution of the number of components in random mappings and partitions
- Symmetric linear spaces of graphs
- On automorphisms of commutative Moufang loops
- On the mixing properties of operations in a finite field
- The predicate method to construct the Post lattice
- Average complexity of searching for identical objects in random nonuniform databases
- On construction of efficient algorithms for solving systems of polynomial Boolean equations by testing a part of variables
