Combinatorial enumeration of groups, graphs, and chemical compounds
Combinatorial enumeration of groups, graphs, and chemical compounds
A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Regular Article: Enumeration of m-Ary Cacti
Advances in Applied Mathematics
The ‘Burnside Process’ Converges Slowly
Combinatorics, Probability and Computing
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Generating Outerplanar Graphs Uniformly at Random
Combinatorics, Probability and Computing
Analytic Combinatorics
Boltzmann samplers for first-order differential specifications
Discrete Applied Mathematics
Boltzmann samplers for v-balanced cycles
Theoretical Computer Science
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We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an “unbiased” way so that a structure of size $n$ gives rise to $n$ pointed structures. We extend Pólya theory to the corresponding pointing operator and present a random sampling framework based on both the principles of Boltzmann sampling and Pólya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated in several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.