 CycleIndexPolynomial - Maple Help

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GroupTheory

 CycleIndexPolynomial
 return the cycle index polynomial of a permutation group 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 ${x}_{1}$, ${x}_{2}$, ..., ${x}_{d}$ 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. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({2}{,}{3}{,}{4}\right)⟩$ (1)
 > $\mathrm{CycleIndexPolynomial}\left(G,\left[a,b,c,d\right]\right)$
 $\frac{{1}}{{24}}{}{{a}}^{{4}}{+}\frac{{1}}{{4}}{}{{a}}^{{2}}{}{b}{+}\frac{{1}}{{3}}{}{a}{}{c}{+}\frac{{1}}{{8}}{}{{b}}^{{2}}{+}\frac{{1}}{{4}}{}{d}$ (2)
 > $\mathrm{CycleIndexPolynomial}\left(\mathrm{CyclicGroup}\left(10\right),\left[x||\left(1..10\right)\right]\right)$
 $\frac{{{\mathrm{x1}}}^{{10}}}{{10}}{+}\frac{{{\mathrm{x2}}}^{{5}}}{{10}}{+}\frac{{2}{}{{\mathrm{x5}}}^{{2}}}{{5}}{+}\frac{{2}{}{\mathrm{x10}}}{{5}}$ (3)
 > $\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(7\right),\left[x||\left(1..7\right)\right]\right)$
 $\frac{{1}}{{14}}{}{{\mathrm{x1}}}^{{7}}{+}\frac{{1}}{{2}}{}{\mathrm{x1}}{}{{\mathrm{x2}}}^{{3}}{+}\frac{{3}}{{7}}{}{\mathrm{x7}}$ (4)
 > $G≔\mathrm{DihedralGroup}\left(7\right)$
 ${G}{≔}{{\mathbf{D}}}_{{7}}$ (5)
 > $E≔\left[\mathrm{op}\right]\left(\mathrm{Elements}\left(G\right)\right)$
 ${E}{≔}\left[\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right){,}\left({1}{,}{6}{,}{4}{,}{2}{,}{7}{,}{5}{,}{3}\right){,}\left({1}{,}{3}\right)\left({4}{,}{7}\right)\left({5}{,}{6}\right){,}\left({1}{,}{5}{,}{2}{,}{6}{,}{3}{,}{7}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{7}\right){,}\left({1}{,}{4}{,}{7}{,}{3}{,}{6}{,}{2}{,}{5}\right){,}\left({1}{,}{5}\right)\left({2}{,}{4}\right)\left({6}{,}{7}\right){,}\left({1}{,}{3}{,}{5}{,}{7}{,}{2}{,}{4}{,}{6}\right){,}\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({1}{,}{7}{,}{6}{,}{5}{,}{4}{,}{3}{,}{2}\right){,}\left({1}{,}{2}\right)\left({3}{,}{7}\right)\left({4}{,}{6}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right){,}\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{4}\right){,}\left(\right)\right]$ (6)
 > $\mathrm{CT}≔\mathrm{sort}\left(\mathrm{map}\left(\mathrm{PermCycleType},E\right)\right)$
 ${\mathrm{CT}}{≔}\left[\left[\right]{,}\left[{7}\right]{,}\left[{7}\right]{,}\left[{7}\right]{,}\left[{7}\right]{,}\left[{7}\right]{,}\left[{7}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}\left[{2}{,}{2}{,}{2}\right]\right]$ (7)

If the vertices of a hexagon are colored with three different colors, then the number of distinct colored hexagons can be calculated by evaluating the cycle index polynomial of the dihedral group of degree $6$ (the group of symmetries of a hexagon) with each indeterminate equal to $3$.

 > $p≔\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(6\right),\left[x||\left(1..6\right)\right]\right)$
 ${p}{≔}\frac{{1}}{{12}}{}{{\mathrm{x1}}}^{{6}}{+}\frac{{1}}{{4}}{}{{\mathrm{x1}}}^{{2}}{}{{\mathrm{x2}}}^{{2}}{+}\frac{{1}}{{3}}{}{{\mathrm{x2}}}^{{3}}{+}\frac{{1}}{{6}}{}{{\mathrm{x3}}}^{{2}}{+}\frac{{1}}{{6}}{}{\mathrm{x6}}$ (8)
 > $\mathrm{eval}\left(p,\left[\mathrm{x1}=3,\mathrm{x2}=3,\mathrm{x3}=3,\mathrm{x4}=3,\mathrm{x5}=3,\mathrm{x6}=3\right]\right)$
 ${92}$ (9)

As a shortcut, you can use the following calling sequence.

 > $\mathrm{CycleIndexPolynomial}\left(\mathrm{DihedralGroup}\left(6\right),\left[3\$6\right]\right)$
 ${92}$ (10) Compatibility

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