The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927. In 1937 it was independently rediscovered by George Pólya, who then greatly popularized the result by applying it to many counting problems, in particular to the enumeration of chemical compounds.
The Pólya enumeration theorem has been incorporated into symbolic combinatorics and the theory of combinatorial species.
Let X be a finite set and let G be a group of permutations of X (or a finite symmetry group that acts on X). The set X may represent a finite set of beads, and G may be a chosen group of permutations of the beads. For example, if X is a necklace of n beads in a circle, then rotational symmetry is relevant so G is the cyclic group Cn, while if X is a bracelet of n beads in a circle, rotations and reflections are relevant so G is the dihedral group Dn of order 2n. Suppose further that Y is a finite set of colors — the colors of the beads — so that YX is the set of colored arrangements of beads (more formally: YX is the set of functions .) Then the group G acts on YX. The Pólya enumeration theorem counts the number of orbits under G of colored arrangements of beads by the following formula:
where is the number of colors and c(g) is the number of cycles of the group element g when considered as a permutation of X.
In the more general and more important version of the theorem, the colors are also weighted in one or more ways, and there could be an infinite number of colors provided that the set of colors has a generating function with finite coefficients. In the univariate case, suppose that
is the generating function of the set of colors, so that there are fw colors of weight w for each integer w ≥ 0. In the multivariate case, the weight of each color is a vector of integers and there is a generating function f(t1, t2, .
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration. Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, ...
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.
Previously, G. Cornuejols and M. Dawande (1998) proposed a set of 0-1 linear programming instances that proved to be very hard to solve by traditional methods, and in particular by linear programming based branch-and-bound. They offered these market split ...