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GroupTheory

  

CycleIndexPolynomial

  

return the cycle index polynomial of a permutation group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

CycleIndexPolynomial( G, vars )

Parameters

G

-

a permutation group

vars

-

list of names

Description

• 

The cycle index polynomial of a permutation group G encodes, in concise form, the cycle structure of the elements of G.  It is the "average" of the cycle index polynomials of the elements of G.

• 

For a permutation p of degree d, the cycle index polynomial in the variables x1, x2, ..., xd is the monomial x1c1x2c2...xdcd, where, for each i, ci is the number of cycles of length i in p.

• 

The CycleIndexPolynomial( G, vars ) command computes the cycle index polynomial of a permutation group G with respect to the variables in the list vars of names.

Examples

withGroupTheory:

GGroupPerm1,2,Perm2,3,4

G1,2,2,3,4

(1)

CycleIndexPolynomialG,a,b,c,d

124a4+14a2b+13ac+18b2+14d

(2)

CycleIndexPolynomialCyclicGroup10,x1..10

x11010+x2510+2x525+2x105

(3)

CycleIndexPolynomialDihedralGroup7,x1..7

114x17+12x1x23+37x7

(4)

GDihedralGroup7

GD7

(5)

EopElementsG

E1,3,5,7,2,4,6,1,72,63,5,2,73,64,5,1,7,6,5,4,3,2,1,23,74,6,1,6,4,2,7,5,3,1,34,75,6,1,5,2,6,3,7,4,1,42,35,7,1,4,7,3,6,2,5,1,52,46,7,1,2,3,4,5,6,7,1,62,53,4,

(6)

CTsortmapPermCycleType,E

CT,7,7,7,7,7,7,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2

(7)

Compatibility

• 

The GroupTheory[CycleIndexPolynomial] command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

GroupTheory

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[Elements]

GroupTheory[PermCycleType]

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