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Simulating the component counts of combinatorial structures.

Richard Arratia1, A D Barbour2, W J Ewens3

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Summary
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This study introduces fast simulation methods for random combinatorial structures like permutations and mappings. These techniques improve efficiency for analyzing component counts in logarithmic assemblies.

Keywords:
Chinese restaurant processEwens sampling formulaFeller couplingRandom mappingsRandom permutationsSpaghetti loop distribution

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Area of Science:

  • Combinatorics
  • Probability Theory
  • Computer Science

Background:

  • Analyzing random combinatorial structures is crucial in various scientific fields.
  • Efficient simulation methods are needed for complex structures like permutations and mappings.
  • Logarithmic combinatorial structures present unique simulation challenges.

Purpose of the Study:

  • To describe and compare simulation methods for component counts in random logarithmic combinatorial structures.
  • To develop a fast and generalizable simulation approach.
  • To analyze specific properties of random mappings and general logarithmic structures.

Main Methods:

  • Exploitation of the Feller coupling for simulating permutations and logarithmic assemblies.
  • Development of an acceptance/rejection method for logarithmic multisets and selections.
  • Utilizing a conditioning relationship representing distributions as conditioned independent random variables.
  • Improving the acceptance rate of the acceptance/rejection method.

Main Results:

  • A very fast method for simulating logarithmic assemblies is presented.
  • An improved acceptance/rejection method is detailed for specific structures.
  • The probability of a random mapping having no repeated component sizes is estimated.
  • The asymptotic distribution for the difference between component and distinct component sizes is established.

Conclusions:

  • The proposed methods offer significant speed improvements for simulating logarithmic combinatorial structures.
  • The techniques provide valuable tools for analyzing the properties of these structures.
  • This work advances the understanding of random mappings and general logarithmic assemblies.