A Cycle Index Spreadsheet

group

S

n

A

n

C

n

D

n

E

n

permutation length n

2

permutation

cycles

cycle length matrix

monomial representation

{1,2}

(1) (2)

cycle length	2	1
count	0	2

2

x

1

{2,1}

(2, 1)

cycle length	2	1
count	1	0

x

2

cycle index

1

2



2

x

1

+

x

2



The cycle index of a permutation group is the mean of a list of monomials, one for each permutation in the permutation group. The monomials are themselves each products of an indexed variable—here

x

—where the indices run from 1 to the length of the permutation. Each variable in each monomial is raised to a certain power ranging from 0 to the length of the permutation. That power is the number of cycles in the permutation that have a length equal to the index. By way of example, the permutation {1,3,2,4} has cycles (1)(2,3)(4). There are thus 2 cycles of length 1 and 1 cycle of length 2. The corresponding monomial is thus

2

x

1

x

2

0

x

3

0

x

4

, which is simplified to

2

x

1

x

2

. This Demonstration shows how the cycle index is computed in a "spreadsheet" sort of a way. You select the permutation group. You then select the length of the permutation. For each permutation, the Demonstration shows the cycle structure, the number of cycles of each possible length and the corresponding monomial. The bottom-right element of the grid shows the mean of the monomials, which is the cycle index.