Let $n$ be a positive integer. A 3-constellation of degree $n$ is a triplet $\left(g_0\,g_1\,g_{\mathrm{\infty }}\right)$ of elements of $S_n$that generate a transitive subgroup of $S_n$ and satisfy $g_0\cdot g_1\cdot g_{\mathrm{\infty }}\=1$.

Two $3$-constellations $\left(g_0\,g_1\,g_{\mathrm{\infty }}\right)$, $\left(h_0\,h_1\,h_{\mathrm{\infty }}\right)$ are conjugated if there exists $\mathrm{\tau }$ in $S_n$ with $\mathrm{\tau }\cdot g_i\cdot {\mathrm{\tau }}^{\mathrm{-1}}\=h_i$ for each $i$ in $\left\{0\,1\,\mathrm{\infty }\right\}$ (or, equivalently, each $i$ in $\left\{0\,1\right\}$).

The branch pattern of $\left(g_0\,g_1\,g_{\mathrm{\infty }}\right)$ is a triplet $\left(B_0\,B_1\,B_{\mathrm{\infty }}\right)$ where $B_i$ is a partition of $n$ giving the cycle-structure of $g_i$. We include $1$-cycles so FindDessins can find $n$ by taking the sum of the entries of each $B_i$.

Given B0, B1, Binf as input, FindDessins computes one representative from every conjugacy class of $3$-constellations with branch pattern (B0, B1, Binf). Each $3$-constellation $\left(g_0\,g_1\,g_{\mathrm{\infty }}\right)$ will be represented by the list $\left[g_0\,g_1\right]$, since $g_{\mathrm{\infty }}$ can be computed as ${\left(g_0\cdot g_1\right)}^{\mathrm{-1}}$.

A conjugacy class of $3$-constellations corresponds 1-1 with a dessin d'enfant, as well as with a Belyi map (up to equivalence). A Belyi map is a holomorphic function from a compact Riemann surface to the Riemann sphere that only ramifies above {0,1,infinity}. So we can count how many dessins, or how many Belyi maps, exists for a given branch pattern by counting the output of FindDessins.

FindDessins implements the strategy of Section $4$ in arXiv:1604.08158 with a number of additions. Progress is reported during the computation by setting infolevel['FindDessins'] to 1 or 2.

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

$B≔\left[1\,\mathrm{`\$`}\left(3\,9\right)\right],\left[\mathrm{`\$`}\left(2\,14\right)\right],\left[\mathrm{`\$`}\left(7\,4\right)\right]$

$B≔\left[1\,3\,3\,3\,3\,3\,3\,3\,3\,3\right],\left[2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\,2\right],\left[7\,7\,7\,7\right]$

$S≔\mathrm{FindDessins}\left(\left[1\,\mathrm{`\$`}\left(3\,9\right)\right]\,\left[\mathrm{`\$`}\left(2\,14\right)\right]\,\left[\mathrm{`\$`}\left(7\,4\right)\right]\right)\:$$\mathrm{NumberOfDessins}≔\mathrm{nops}\left(S\right)$

$\mathrm{NumberOfDessins}≔1$

$d≔S\left[1\right]$

$d≔\left[\left(2\,3\,4\right)\left(5\,6\,7\right)\left(8\,9\,10\right)\left(11\,12\,13\right)\left(14\,15\,16\right)\left(17\,18\,19\right)\left(20\,21\,22\right)\left(23\,24\,25\right)\left(26\,27\,28\right)\,\left(1\,2\right)\left(3\,5\right)\left(4\,8\right)\left(6\,11\right)\left(7\,14\right)\left(9\,17\right)\left(10\,20\right)\left(12\,22\right)\left(13\,16\right)\left(15\,23\right)\left(18\,21\right)\left(19\,26\right)\left(24\,27\right)\left(25\,28\right)\right]$

$\mathrm{g0}≔d\left[1\right]$

$\mathrm{g0}≔\left(2\,3\,4\right)\left(5\,6\,7\right)\left(8\,9\,10\right)\left(11\,12\,13\right)\left(14\,15\,16\right)\left(17\,18\,19\right)\left(20\,21\,22\right)\left(23\,24\,25\right)\left(26\,27\,28\right)$

$\mathrm{g1}≔d\left[2\right]$

$\mathrm{g1}≔\left(1\,2\right)\left(3\,5\right)\left(4\,8\right)\left(6\,11\right)\left(7\,14\right)\left(9\,17\right)\left(10\,20\right)\left(12\,22\right)\left(13\,16\right)\left(15\,23\right)\left(18\,21\right)\left(19\,26\right)\left(24\,27\right)\left(25\,28\right)$

$g\left[\mathrm{\infty }\right]≔{\left(\mathrm{g0}·\mathrm{g1}\right)}^{-1}$

$g_{\mathrm{\infty }}≔\left(1\,4\,10\,22\,11\,5\,2\right)\left(3\,7\,16\,12\,21\,17\,8\right)\left(6\,13\,15\,25\,27\,23\,14\right)\left(9\,19\,28\,24\,26\,18\,20\right)$

$\mathrm{DecomposeDessin}\left(d\right)$

$''indecomposable''$

The GroupTheory[FindDessins] and GroupTheory[DecomposeDessin] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.