Cycle index polynomial of a permutation group in given variables
CycleIndexPolynomial[perm, {x1, …, xn}]
CycleIndexPolynomial[group, {x1, …, xn}]
CycleIndexPolynomial[AlternatingGroup[5], {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4], Subscript[x, 5]}]
→ Subscript[x, 1]^5/60 + (Subscript[x, 1]*Subscript[x, 2]^2)/4 + (Subscript[x, 1]^2*Subscript[x, 3])/3 + (2*Subscript[x, 5])/5CycleIndexPolynomial[DihedralGroup[8], Array[Subscript[x, ##]&, 8]]
→ Subscript[x, 1]^8/16 + (5*Subscript[x, 2]^4)/16 + Subscript[x, 4]^2/8 + Subscript[x, 8]/4 + (Subscript[x, 1]^2*Subscript[x, 2]^3)/4CycleIndexPolynomial[PermutationGroup[{Cycles[{{1, 2}}], Cycles[{{1, 3}, {2, 4}}]}], {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4]}]
→ Subscript[x, 1]^4/8 + (3*Subscript[x, 2]^2)/8 + (Subscript[x, 1]^2*Subscript[x, 2])/4 + Subscript[x, 4]/4