9.2 Cycle Index Polynomials

Cycle index polynomials are powerful tools in combinatorics that encode the cycle structure of permutation groups. They're key to solving counting problems involving symmetry, like figuring out how many unique ways you can color a cube's faces.

These polynomials connect to Polya's Enumeration Theorem, which helps count orbits of group actions. By substituting values into cycle index polynomials, we can solve tricky counting problems in chemistry, graph theory, and other fields where symmetry plays a role.

Cycle types and structure

Permutations and cycle representation

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• A permutation is a bijective function from a set to itself
• Permutations can be represented as a product of disjoint cycles
  • Each cycle corresponds to a subset of elements that are cyclically permuted by the permutation
  • Elements not included in any cycle are fixed points of the permutation
• The cyclic structure of a permutation determines its properties and behavior

Cycle types and conjugacy classes

• The cycle type of a permutation is a partition of the size of the set it acts on
  • Each part of the partition corresponds to the length of a cycle in the permutation's disjoint cycle representation
  • Example: A permutation with cycle type (3, 2, 1) has one 3-cycle, one 2-cycle, and one fixed point
• The cycle type determines the conjugacy class of a permutation
  • Two permutations are conjugate if and only if they have the same cycle type
  • Conjugate permutations have the same cyclic structure and properties
• The number of permutations with a given cycle type can be calculated using the multinomial coefficient formula
  • The formula accounts for the number of ways to arrange elements into cycles of the specified lengths

Cycle index polynomials

Definition and construction

• The cycle index polynomial of a permutation group G is a polynomial in variables $s_1, s_2, \ldots, s_n$, where $n$ is the size of the set on which G acts
• Each term of the cycle index polynomial corresponds to a conjugacy class of G
  • The coefficient of each term is the number of permutations in that class divided by the order of G
  • The exponent of each variable $s_i$ in a term is the number of $i$-cycles in the cycle type of the permutations in the corresponding conjugacy class
• The cycle index polynomial encodes information about the cycle structure of all permutations in the group

Properties and operations

• The cycle index polynomial of a direct product of permutation groups is the product of their individual cycle index polynomials
  • This property allows for the computation of cycle index polynomials of larger groups from those of smaller ones
• The cycle index polynomial of a wreath product of permutation groups can be expressed in terms of the cycle index polynomials of the component groups
• The cycle index polynomial of a permutation group acting on a set of functions or configurations can be obtained by substituting appropriate generating functions for the variables $s_i$

Enumeration problems with polynomials

Counting orbits of group actions

• The cycle index polynomial can be used to count the number of orbits of a permutation group acting on a set of functions or configurations
• To count the number of orbits of a group G acting on colorings of a set with $m$ colors, substitute $s_i = m^i$ in the cycle index polynomial of G
  • The resulting polynomial, evaluated at $m$, gives the number of orbits, which is the number of distinct colorings up to the action of G
  • Example: The number of distinct ways to color the vertices of a square with $m$ colors, up to rotations, is given by $\frac{1}{4}(m^4 + 2m^2 + m)$
• This technique can be extended to count the number of orbits of G acting on other types of configurations, such as graphs or chemical compounds, by using appropriate generating functions in place of the variables $s_i$

Applications and examples

• Polya's enumeration theorem, which relates the cycle index polynomial of a group to the number of orbits of its action on a set of functions, has numerous applications in combinatorics and discrete mathematics
• Examples of problems that can be solved using cycle index polynomials include:
  • Counting the number of distinct necklaces with a given number of beads and colors, up to rotations and reflections
  • Enumerating the number of non-isomorphic graphs with a given number of vertices and edges, up to relabeling of vertices
  • Determining the number of distinct chemical compounds with a given molecular formula, up to permutations of identical atoms

Polynomials vs generating functions

Relationship between cycle index polynomials and generating functions

• Cycle index polynomials are a generalization of generating functions
• Generating functions can be obtained from cycle index polynomials by substituting $s_i = x^i$ in the cycle index polynomial
  • The resulting generating function encodes the number of permutations in the group with a given number of fixed points
  • Example: The generating function for the number of permutations in $S_n$ with $k$ fixed points is obtained by substituting $s_i = x^i$ in the cycle index polynomial of $S_n$ and extracting the coefficient of $x^k$
• The exponential generating function for the number of permutations in the symmetric group $S_n$ can be obtained from the cycle index polynomial of $S_n$ by substituting $s_i = \frac{x^i}{i!}$
  • This generating function is given by $\exp(x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots)$
• The ordinary generating function for the number of permutations in $S_n$ with a given number of cycles can be obtained from the cycle index polynomial of $S_n$ by substituting $s_i = x^i$

Identities and formulas

• The relationships between cycle index polynomials and generating functions allow for the derivation of various identities and formulas involving permutations and their cycle structure
• Examples of such identities include:
  • The exponential formula, which relates the exponential generating function of a sequence to the ordinary generating function of its cycle index polynomials
  • The cycle index polynomial of the symmetric group $S_n$ can be expressed as a determinant involving the Stirling numbers of the first kind
  • The generating function for the number of permutations in $S_n$ with a given number of cycles satisfies a recurrence relation involving the Stirling numbers of the first kind
• These identities provide insights into the structure and properties of permutations and their cycle types, and can be used to derive new results and solve problems in combinatorics and related fields