Boltzmann Samplers, Pólya Theory, and Cycle Pointing

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 Polya theory to the corresponding pointing operator and present a random sampling framework based on both the principles of Boltzmann sampling and Polya 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.