# Logarithmic Combinatorial Structures: A Probabilistic Approach

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*Richard Arratia; A. D. Barbour; Simon Tavaré*

A publication of the European Mathematical Society

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 logarithmically with
the size of the element, and the size of the largest component is an
appreciable fraction of the whole.

This book explains the similarities in asymptotic behavior as the result of
two basic properties shared by the structures: the conditioning relation and
the logarithmic condition. The discussion is conducted in the language of
probability, enabling the theory to be developed under rather general and
explicit conditions; for the finer conclusions, Stein's method emerges as the
key ingredient. The book is thus of particular interest to graduate students
and researchers in both combinatorics and probability theory.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in probability theory and stochastic processes.