partitions

n = 20;
Manipulate[
f[l_] := Accumulate[Join[{{0, 0}}, MapIndexed[#1 (
       With[{i = #2[[1]]},
        N[{Sin[2 Pi i/k], Cos[2 Pi i / k]}]
        ]
       ) &, l]]];

partitions = IntegerPartitions[n];
paths = f /@ partitions;
Graphics[
 {
  Thick,
  Opacity[0.5],
  PointSize[Large],
  Table[
   {
    ColorData["DarkRainbow"][i/Length[paths]],
    Line[paths[[i]]],
    Point[paths[[i, -1]]]
    },
   {i, Length[paths]}
   ]
  }
 ],
{k, 3, 20, 1}
]