As for generating the notations automatically, that's exactly what I was looking for here . But then, Marek answered by enumerating the list up to 10D. I'd still be interested in over 10D, where at 11D the list is six thousand strong. What would be really neat, is a notation --> equation converter. When I decide to drop some $$$ on a nice fast computer, it'll help with big toratopes.
Toratopes[1] = {1};
Rotatopes[n_] := Rest[IntegerPartitions[n]]
ExpandRotatope[list_] :=
DeleteDuplicates[
Map[Reverse @* Sort, Tuples[Map[Toratopes, list]], 1]]
Toratopes[n_] := Flatten[Map[ExpandRotatope, Rotatopes[n]], 1]
TStringList[1] = "I";
TStringList[list_] :=
Flatten[{"(", Table[TStringList[i], {i, list}], ")"}, 1]
ClosedToratopeNotation[list_] := StringJoin[TStringList[list]]
OpenToratopeNotation[list_] :=
StringJoin[ TStringList[list] // Most // Rest]
ToratopeList[n_] :=
If[n === 1, {"I"},
Join[Map[OpenToratopeNotation, Toratopes[n]],
Map[ClosedToratopeNotation, Toratopes[n]]]] // TableForm
ToratopeList[7]
IIIIIII
(II)IIIII
(II)(II)III
(II)(II)(II)I
((II)I)IIII
(III)IIII
((II)I)(II)II
(III)(II)II
((II)I)(II)(II)
(III)(II)(II)
((II)I)((II)I)I
(III)((II)I)I
(III)(III)I
((III)I)III
(((II)I)I)III
((II)(II))III
((II)II)III
(IIII)III
((III)I)(II)I
(((II)I)I)(II)I
((II)(II))(II)I
((II)II)(II)I
(IIII)(II)I
((III)I)((II)I)
(III)((III)I)
(((II)I)I)((II)I)
(III)(((II)I)I)
((II)(II))((II)I)
(III)((II)(II))
((II)II)((II)I)
((II)II)(III)
(IIII)((II)I)
(IIII)(III)
((IIII)I)II
(((II)II)I)II
(((II)(II))I)II
((((II)I)I)I)II
(((III)I)I)II
((III)(II))II
(((II)I)(II))II
((III)II)II
(((II)I)II)II
((II)(II)I)II
((II)III)II
(IIIII)II
((IIII)I)(II)
(((II)II)I)(II)
(((II)(II))I)(II)
((((II)I)I)I)(II)
(((III)I)I)(II)
((III)(II))(II)
(((II)I)(II))(II)
((III)II)(II)
(((II)I)II)(II)
((II)(II)I)(II)
((II)III)(II)
(IIIII)(II)
((IIIII)I)I
(((II)III)I)I
(((II)(II)I)I)I
((((II)I)II)I)I
(((III)II)I)I
((((II)I)(II))I)I
(((III)(II))I)I
((((III)I)I)I)I
(((((II)I)I)I)I)I
((((II)(II))I)I)I
((((II)II)I)I)I
(((IIII)I)I)I
((IIII)(II))I
(((II)II)(II))I
(((II)(II))(II))I
((((II)I)I)(II))I
(((III)I)(II))I
((IIII)II)I
(((II)II)II)I
(((II)(II))II)I
((((II)I)I)II)I
(((III)I)II)I
((III)(III))I
((III)((II)I))I
(((II)I)((II)I))I
((III)(II)I)I
(((II)I)(II)I)I
((III)III)I
(((II)I)III)I
((II)(II)(II))I
((II)(II)II)I
((II)IIII)I
(IIIIII)I
(IIIIIII)
((II)IIIII)
((II)(II)III)
((II)(II)(II)I)
(((II)I)IIII)
((III)IIII)
(((II)I)(II)II)
((III)(II)II)
(((II)I)(II)(II))
((III)(II)(II))
(((II)I)((II)I)I)
((III)((II)I)I)
((III)(III)I)
(((III)I)III)
((((II)I)I)III)
(((II)(II))III)
(((II)II)III)
((IIII)III)
(((III)I)(II)I)
((((II)I)I)(II)I)
(((II)(II))(II)I)
(((II)II)(II)I)
((IIII)(II)I)
(((III)I)((II)I))
((III)((III)I))
((((II)I)I)((II)I))
((III)(((II)I)I))
(((II)(II))((II)I))
((III)((II)(II)))
(((II)II)((II)I))
(((II)II)(III))
((IIII)((II)I))
((IIII)(III))
(((IIII)I)II)
((((II)II)I)II)
((((II)(II))I)II)
(((((II)I)I)I)II)
((((III)I)I)II)
(((III)(II))II)
((((II)I)(II))II)
(((III)II)II)
((((II)I)II)II)
(((II)(II)I)II)
(((II)III)II)
((IIIII)II)
(((IIII)I)(II))
((((II)II)I)(II))
((((II)(II))I)(II))
(((((II)I)I)I)(II))
((((III)I)I)(II))
(((III)(II))(II))
((((II)I)(II))(II))
(((III)II)(II))
((((II)I)II)(II))
(((II)(II)I)(II))
(((II)III)(II))
((IIIII)(II))
(((IIIII)I)I)
((((II)III)I)I)
((((II)(II)I)I)I)
(((((II)I)II)I)I)
((((III)II)I)I)
(((((II)I)(II))I)I)
((((III)(II))I)I)
(((((III)I)I)I)I)
((((((II)I)I)I)I)I)
(((((II)(II))I)I)I)
(((((II)II)I)I)I)
((((IIII)I)I)I)
(((IIII)(II))I)
((((II)II)(II))I)
((((II)(II))(II))I)
(((((II)I)I)(II))I)
((((III)I)(II))I)
(((IIII)II)I)
((((II)II)II)I)
((((II)(II))II)I)
(((((II)I)I)II)I)
((((III)I)II)I)
(((III)(III))I)
(((III)((II)I))I)
((((II)I)((II)I))I)
(((III)(II)I)I)
((((II)I)(II)I)I)
(((III)III)I)
((((II)I)III)I)
(((II)(II)(II))I)
(((II)(II)II)I)
(((II)IIII)I)
((IIIIII)I)
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