Cycle Index

The cycle index of a permutation group G is the average of the cycle index monomials of all the permutations g in G.

More formally, let G be a permutation group of order m and degree n. Every permutation g in G has a unique decomposition into disjoint cycles, say c₁ c₂ c₃ ... Let the length of a cycle c be denoted by |c|.

Now let j_k(g) be the number of cycles of g of length k, where

$0 \le j_k(g) \le \lfloor n/k \rfloor \mbox{ and } \sum_{k=1}^n k \, j_k(g) \; = n.$

We associate to g the monomial

in the variables a₁, a₂, ... a_n.

Then the cycle index Z(G) of G is given by

Read more about Cycle Index: Example, Types of Actions, Applications

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