Combinatorics | Enumerative combinatorics
Cycle index
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, … that describes the cycle type of this partition: the exponent of ai is the number of cycles of π of size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index. Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at the core of species theory. (Wikipedia).
