Boltzmann Samplers, Pólya Theory, and Cycle Pointing

Abstract

Abstract. 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 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 on Pólya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on 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. This is the extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sam-

The continuum random tree is the scaling limit of unlabeled unrooted trees

“…In the case , Benjamini‐Schramm convergence for was independently obtained by Georgakopoulos and Wagner using different techniques. Our methods for the proof of Theorems and are based on the cycle pointing decomposition established recently by Bodirsky, Fusy, Kang and Vigerske . This novel and effective centering method differs fundamentally from classical approaches, such as the geometric center, and applies to arbitrary classes of combinatorial structures.…”

“…More details on this structural result are given in [, Section 4.3], [, Sec. 2.6.2] and [, Section 3].…”

“…The cycle pointing operator was constructed in and maps a species to the species such that the ‐objects over a set U are pairs ( G , τ ) with and τ a marked cycle of an arbitrary automorphism of G . Here we count fixed points as 1‐cycles.…”

“…Cycle pointed species come with a set of new operations introduced in . If is a cycle‐pointed species and a species, then the pointed product is the subspecies of given by all cycle‐pointed objects such that the marked cycle consists of atoms of the ‐structure and the ‐structure together with this cycle belongs to .…”

“…The Pólya‐Boltzmann model was introduced in : Suppose that we are given a sequence of real numbers s 1 , s 2 ,… ≥ 0 such that . Then we may consider the probability distribution on the set that assigns the probability weight for each n and symmetry .…”