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 $x_1$, $x_2$, ..., $x_d$ is the monomial $x_1^{c_1}{x}_2^{c_2}\mathrm{...}{x}_d^{c_d}$, where, for each $i$, $c_i$ 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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,3\,4\right]\right]\right)\right)$

$G≔⟨\left(1\,2\right)\,\left(2\,3\,4\right)⟩$

$\mathrm{CycleIndexPolynomial}\left(G\,\left[a\,b\,c\,d\right]\right)$

$\frac{1}{24}{a}^4+\frac{1}{4}{a}^{2}b+\frac{1}{3}ac+\frac{1}{8}{b}^2+\frac{1}{4}d$

$\mathrm{CycleIndexPolynomial}\left(\mathrm{CyclicGroup}\left(10\right)\,\left[x\Vert \left(1..10\right)\right]\right)$

$\frac{{\mathrm{x1}}^{10}}{10}+\frac{{\mathrm{x2}}^5}{10}+\frac{2{\mathrm{x5}}^2}{5}+\frac{2\mathrm{x10}}{5}$

$\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(7\right)\,\left[x\Vert \left(1..7\right)\right]\right)$

$\frac{1}{14}{\mathrm{x1}}^7+\frac{1}{2}\mathrm{x1}{\mathrm{x2}}^3+\frac{3}{7}\mathrm{x7}$

$G≔\mathrm{DihedralGroup}\left(7\right)$

$G≔{\mathbf{D}}_7$

$E≔\left[\mathrm{op}\right]\left(\mathrm{Elements}\left(G\right)\right)$

$E≔\left[\left(1\,5\right)\left(2\,4\right)\left(6\,7\right)\,\left(1\,6\,4\,2\,7\,5\,3\right)\,\left(1\,7\,6\,5\,4\,3\,2\right)\,\left(1\,7\right)\left(2\,6\right)\left(3\,5\right)\,\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\,\left(1\,2\right)\left(3\,7\right)\left(4\,6\right)\,\left(1\,3\,5\,7\,2\,4\,6\right)\,\left(1\,3\right)\left(4\,7\right)\left(5\,6\right)\,\left(1\,4\,7\,3\,6\,2\,5\right)\,\left(1\,4\right)\left(2\,3\right)\left(5\,7\right)\,\left(1\,5\,2\,6\,3\,7\,4\right)\,\left(1\,2\,3\,4\,5\,6\,7\right)\,\left(1\,6\right)\left(2\,5\right)\left(3\,4\right)\,\left(\right)\right]$

$\mathrm{CT}≔\mathrm{sort}\left(\mathrm{map}\left(\mathrm{PermCycleType}\,E\right)\right)$

$\mathrm{CT}≔\left[\left[\right]\,\left[7\right]\,\left[7\right]\,\left[7\right]\,\left[7\right]\,\left[7\right]\,\left[7\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\,\left[2\,2\,2\right]\right]$

$p≔\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(6\right)\,\left[x\Vert \left(1..6\right)\right]\right)$

$p≔\frac{1}{12}{\mathrm{x1}}^6+\frac{1}{4}{\mathrm{x1}}^2{\mathrm{x2}}^2+\frac{1}{3}{\mathrm{x2}}^3+\frac{1}{6}{\mathrm{x3}}^2+\frac{1}{6}\mathrm{x6}$

$\mathrm{eval}\left(p\,\left[\mathrm{x1}=3\,\mathrm{x2}=3\,\mathrm{x3}=3\,\mathrm{x4}=3\,\mathrm{x5}=3\,\mathrm{x6}=3\right]\right)$

$92$

$\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(6\right)\,\left[\mathrm{`\$`}\left(3\,6\right)\right]\right)$

$92$

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

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