# Logarithmic Combinatorial Structures: A Probabilistic Approach

@inproceedings{Arratia2003LogarithmicCS, title={Logarithmic Combinatorial Structures: A Probabilistic Approach}, author={Richard Arratia and Andrew D. Barbour and Simon Tavar{\'e}}, year={2003} }

The elements of many classical combinatorial structures can be naturally decomposed into components. Permutations can be decomposed into cycles, polynomials over a finite field into irreducible factors, mappings into connected components. In all of these examples, and in many more, there are strong similarities between the numbers of components of different sizes that are found in the decompositions of `typical' elements of large size. For instance, the total number of components grows…

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## References

SHOWING 1-10 OF 193 REFERENCES

### Random Combinatorial Structures and Prime Factorizations

- Mathematics
- 1997

Introduction Many combinatorial structures decompose into components, with the list of component sizes carrying substantial information. An integer factors into primes—this is a similar situation,…

### Independent Process Approximations for Random Combinatorial Structures

- Mathematics
- 1994

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted…

### Gaussian limiting distributions for the number of components in combinatorial structures

- MathematicsJ. Comb. Theory, Ser. A
- 1990

### On random polynomials over finite fields

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1993

Abstract We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally,…

### Combinatorial species and tree-like structures

- Computer ScienceEncyclopedia of mathematics and its applications
- 1997

The combinatorial theory of species, introduced by Joyal in 1980, provides a unified understanding of the use of generating functions for both labelled and unlabelled structures and as a tool for the…

### Total Variation Asymptotics for Poisson Process Approximations of Logarithmic Combinatorial Assemblies

- Mathematics
- 1995

Assemblies are the decomposable combinatorial constructions characterized by the exponential formula for generating functions : Σp(n)s n /n ! = exp(Σm i s i /i !). Here p(n) is the total number of…

### Limit Theorems for Combinatorial Structures via Discrete Process Approximations

- MathematicsRandom Struct. Algorithms
- 1992

The power ofrete functional limitTheorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods are demonstrated.

### Order Statistics for Decomposable Combinatorial Structures

- MathematicsRandom Struct. Algorithms
- 1994

It is shown that when the generating function for the components of a structure is a logarithmic function, then the joint distribution of the normalized order statistics of the component sizes of a random object of size n coverges to the Poisson–Dirichlet distribution on the simplex.

### Exact Largest and Smallest Size of Components

- Mathematics, Computer ScienceAlgorithmica
- 2001

The number of decomposable combinatorial structures with largest and smallest components of a given size is provided for the number of permutations of size n with largest cycle length equal to k.

### On the asymptotic distribution of large prime factors

- Mathematics
- 1993

A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2 > ... ^ aM. We establish the asymptotic distribution, as «-»• oo, of…