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1}}}, {{{1, 1}, {1, 3}}, {{2, 1}, {1, 2}}, {{1, 4}}}, {{{1, 1}, {3, 3}}, {{4, 1}, {3, 2}}, {{1, 2}, {2, 4}}}, {{{2, 1}, {8, 3}}, {{8, 1}, {9, 2}}, {{1, 2}, {6, 4}}}, {{{2, 1}, {19, 3}}, {{13, 1}, {23, 2}}, {{1, 1}, {1, 2}, {14, 4}}}, {{{2, 1}, {46, 3}}, {{24, 1}, {58, 2}}, {{2, 2}, {34, 4}}}, {{{ 4, 1}, {101, 3}}, {{39, 1}, {134, 2}}, {{1, 1}, {3, 2}, {75, 4}}}}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ sn = Sort@Flatten[solidformBTK /@ Partitions[n]]; {Frequencies[ Length /@ ToCycles[\ Ordering[Map[turn, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[Ordering[Map[flip@\ #\ &, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[ Ordering[Map[lapse@\ #\ &, sn, {1}]]]\ ]}, {n, 8, 12}] // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({{{6, 1}, {226, 3}}, {{68, 1}, {308, 2}}, {{6, 2}, {168, 4}}}\)}, {\({{{6, 1}, {486, 3}}, {{110, 1}, {677, 2}}, {{2, 1}, {7, 2}, {362, 4}}}\)}, {\({{{8, 1}, {1038, 3}}, {{182, 1}, {1470, 2}}, 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{14, 4}}}, {{{2, 1}, {46, 3}}, {{24, 1}, {58, 2}}, {{2, 2}, {34, 4}}}, {{{4, 1}, {101, 3}}, {{39, 1}, {134, 2}}, {{1, 1}, {3, 2}, {75, 4}}}, {{{6, 1}, {226, 3}}, {{68, 1}, {308, 2}}, {{6, 2}, {168, 4}}}, {{{6, 1}, {486, 3}}, {{110, 1}, {677, 2}}, {{2, 1}, {7, 2}, {362, 4}}}, {{{8, 1}, {1038, 3}}, {{182, 1}, {1470, 2}}, {{11, 2}, {775, 4}}}, {{{11, 1}, {2163, 3}}, {{288, 1}, {3106, 2}}, {{2, 1}, {15, 2}, {1617, 4}}}, {{{13, 1}, {4471, 3}}, {{468, 1}, {6479, 2}}, {{25, 2}, {3344, 4}}}}\)], "Output"] }, Open ]], Cell["the LL was forgotten:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[sn = Sort@Flatten[solidform /@ Partitions[n]]; temp = Ordering[Map[lapse@\ #\ &, sn, {1}]]; {Frequencies[ Length /@ ToCycles[\ \ temp[\([temp]\)]\ \ ]\ ]}, {n, 7}] // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({{{1, 1}}}\)}, {\({{{2, 2}}}\)}, {\({{{2, 1}, {4, 2}}}\)}, {\({{{2, 1}, {12, 2}}}\)}, {\({{{3, 1}, {28, 2}}}\)}, {\({{{4, 1}, {68, 2}}}\)}, {\({{{7, 1}, {150, 2}}}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {{{{1, 1}}}, {{{2, 2}}}, {{{2, 1}, {4, 2}}}, {{{2, 1}, {12, 2}}}, {{{3, 1}, {28, 2}}}, {{{4, 1}, {68, 2}}}, {{{7, 1}, {150, 2}}}}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[sn = Sort@Flatten[solidformBTK /@ Partitions[n]]; temp = Ordering[Map[lapse@\ #\ &, sn, {1}]]; {Frequencies[ Length /@ ToCycles[\ \ temp[\([temp]\)]\ \ ]\ ]}, {n, 8, 12}] // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\({{{12, 1}, {336, 2}}}\)}, {\({{{16, 1}, {724, 2}}}\)}, {\({{{22, 1}, {1550, 2}}}\)}, {\({{{32, 1}, {3234, 2}}}\)}, {\({{{50, 1}, {6688, 2}}}\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {{{{12, 1}, {336, 2}}}, {{{16, 1}, {724, 2}}}, {{{22, 1}, { 1550, 2}}}, {{{32, 1}, {3234, 2}}}, {{{50, 1}, {6688, 2}}}}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ sn = Sort@Flatten[solidformBTK /@ Partitions[n]]; {Frequencies[ Length /@ ToCycles[\ Ordering[Map[turn, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[Ordering[Map[flip@\ #\ &, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[ temp = Ordering[ Map[lapse@\ #\ &, sn, {1}]]]\ ], \[IndentingNewLine]Frequencies[ Length /@ ToCycles[\ \ temp[\([temp]\)]\ \ ]\ ]\[IndentingNewLine]}, {n, 13, 13}]\)], "Input"], Cell[BoxData[ \({{{{17, 1}, {9077, 3}}, {{728, 1}, {13260, 2}}, {{2, 1}, {33, 2}, {6795, 4}}, {{68, 1}, {13590, 2}}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ sn = Sort@Flatten[solidformBTK /@ Partitions[n]]; {Frequencies[ Length /@ ToCycles[\ Ordering[Map[turn, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[Ordering[Map[flip@\ #\ &, sn, {2}]]]\ ], Frequencies[ Length /@ ToCycles[ temp = Ordering[ Map[lapse@\ #\ &, sn, {1}]]]\ ], \[IndentingNewLine]Frequencies[ Length /@ ToCycles[\ \ temp[\([temp]\)]\ \ ]\ ]\[IndentingNewLine]}, {n, 14, 15}]\)], "Input"], Cell[BoxData[ \({{{{24, 1}, {18260, 3}}, {{1150, 1}, {26827, 2}}, {{48, 2}, {13677, 4}}, {{96, 1}, {27354, 2}}}, {{{28, 1}, {36258, 3}}, {{1770, 1}, {53516, 2}}, {{4, 1}, {65, 2}, {27167, 4}}, {{134, 1}, {54334, 2}}}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["combinations", "Section"], Cell[BoxData[ \(\(s\ = Sort@Flatten[solidform /@ Partitions[4]];\)\)], "Input"], Cell[BoxData[ \(\(i = Range[Length[s]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(cf = ToCycles[f = Ordering[Map[flip@\ #\ &, s, {2}]]]\)], "Input"], Cell[BoxData[ \({{1}, {6, 2}, {7, 3}, {11, 4}, {13, 5}, {8}, {9}, {12, 10}, {14}, {15}, {19, 16}, {20, 17}, {22, 18}, {21}, {23}, {25, 24}, {26}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ct = ToCycles[t\ = Ordering[Map[turn, s, {2}]]]\)], "Input"], Cell[BoxData[ \({{5, 13, 1}, {8, 6, 2}, {10, 11, 3}, {12, 7, 4}, {9}, {16, 19, 14}, {18, 22, 15}, {21, 20, 17}, {24, 25, 23}, {26}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cl = ToCycles[l = Ordering[Map[lapse@\ #\ &, s, {1}]]]\)], "Input"], Cell[BoxData[ \({{5, 13, 26, 1}, {8, 19, 14, 2}, {10, 22, 23, 3}, {12, 25, 15, 4}, {16, 6}, {18, 11, 24, 7}, {21, 20, 17, 9}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(alle = Union@\(Flatten@ Outer[h@ToCycles[\ \(#1[\([#2]\)]\)\ [\([#3]\)]] &, {i, f, t, l}, {i, f, t, l}\ , {i, f, t, l}, 1]\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[%]\)], "Input"], Cell[BoxData[ \(24\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(idx = Outer[Position[alle, h@ToCycles[\ \(#1[\([#2]\)]\)[\([#3]\)]\ ], 1, 1] &, {i, f, t, l}, {i, f, t, l}, {i, f, t, l}, 1] /. \ {{k_Integer}} \[Rule] k\)], "Input"], Cell[BoxData[ \({{{24, 19, 9, 1}, {19, 24, 22, 12}, {9, 21, 11, 4}, {1, 10, 3, 16}}, {{19, 24, 22, 12}, {24, 19, 9, 1}, {22, 11, 21, 15}, {12, 3, 10, 2}}, {{9, 21, 11, 4}, {21, 9, 19, 7}, {11, 22, 24, 18}, {4, 16, 20, 8}}, {{1, 10, 3, 16}, {10, 1, 17, 5}, {3, 12, 23, 14}, {16, 4, 13, 6}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(idx // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {"24"}, {"19"}, {"9"}, {"1"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"19"}, {"24"}, {"22"}, {"12"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"9"}, {"21"}, {"11"}, {"4"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"1"}, {"10"}, {"3"}, {"16"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {"19"}, {"24"}, {"22"}, {"12"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"24"}, {"19"}, {"9"}, {"1"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"22"}, {"11"}, {"21"}, {"15"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"12"}, {"3"}, {"10"}, {"2"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {"9"}, {"21"}, {"11"}, {"4"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"21"}, {"9"}, {"19"}, {"7"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"11"}, {"22"}, {"24"}, {"18"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"4"}, {"16"}, {"20"}, {"8"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {"1"}, {"10"}, {"3"}, {"16"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"10"}, {"1"}, {"17"}, {"5"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"3"}, {"12"}, {"23"}, {"14"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {"16"}, {"4"}, {"13"}, {"6"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{{24, 19, 9, 1}, {19, 24, 22, 12}, {9, 21, 11, 4}, {1, 10, 3, 16}}, {{19, 24, 22, 12}, {24, 19, 9, 1}, {22, 11, 21, 15}, {12, 3, 10, 2}}, {{9, 21, 11, 4}, {21, 9, 19, 7}, {11, 22, 24, 18}, {4, 16, 20, 8}}, {{1, 10, 3, 16}, {10, 1, 17, 5}, {3, 12, 23, 14}, {16, 4, 13, 6}}}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((tag = Outer[q[\ #1, #2, #3\ ] &\ , {ii, ff, tt, ll}, {ii, ff, tt, ll}, {ii, ff, tt, ll}, 1])\) // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(q[ii, ii, ii]\)}, {\(q[ii, ii, ff]\)}, {\(q[ii, ii, tt]\)}, {\(q[ii, ii, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, ff, ii]\)}, {\(q[ii, ff, ff]\)}, {\(q[ii, ff, tt]\)}, {\(q[ii, ff, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, tt, ii]\)}, {\(q[ii, tt, ff]\)}, {\(q[ii, tt, tt]\)}, {\(q[ii, tt, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, ll, ii]\)}, {\(q[ii, ll, ff]\)}, {\(q[ii, ll, tt]\)}, {\(q[ii, ll, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {\(q[ff, ii, ii]\)}, {\(q[ff, ii, ff]\)}, {\(q[ff, ii, tt]\)}, {\(q[ff, ii, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ff, ff, ii]\)}, {\(q[ff, ff, ff]\)}, {\(q[ff, ff, tt]\)}, {\(q[ff, ff, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ff, tt, ii]\)}, {\(q[ff, tt, ff]\)}, {\(q[ff, tt, tt]\)}, {\(q[ff, tt, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ff, ll, ii]\)}, {\(q[ff, ll, ff]\)}, {\(q[ff, ll, tt]\)}, {\(q[ff, ll, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {\(q[tt, ii, ii]\)}, {\(q[tt, ii, ff]\)}, {\(q[tt, ii, tt]\)}, {\(q[tt, ii, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[tt, ff, ii]\)}, {\(q[tt, ff, ff]\)}, {\(q[tt, ff, tt]\)}, {\(q[tt, ff, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[tt, tt, ii]\)}, {\(q[tt, tt, ff]\)}, {\(q[tt, tt, tt]\)}, {\(q[tt, tt, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[tt, ll, ii]\)}, {\(q[tt, ll, ff]\)}, {\(q[tt, ll, tt]\)}, {\(q[tt, ll, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {\(q[ll, ii, ii]\)}, {\(q[ll, ii, ff]\)}, {\(q[ll, ii, tt]\)}, {\(q[ll, ii, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ll, ff, ii]\)}, {\(q[ll, ff, ff]\)}, {\(q[ll, ff, tt]\)}, {\(q[ll, ff, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ll, tt, ii]\)}, {\(q[ll, tt, ff]\)}, {\(q[ll, tt, tt]\)}, {\(q[ll, tt, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ll, ll, ii]\)}, {\(q[ll, ll, ff]\)}, {\(q[ll, ll, tt]\)}, {\(q[ll, ll, ll]\)} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{{ q[ ii, ii, ii], q[ ii, ii, ff], q[ ii, ii, tt], q[ ii, ii, ll]}, { q[ ii, ff, ii], q[ ii, ff, ff], q[ ii, ff, tt], q[ ii, ff, ll]}, { q[ ii, tt, ii], q[ ii, tt, ff], q[ ii, tt, tt], q[ ii, tt, ll]}, { q[ ii, ll, ii], q[ ii, ll, ff], q[ ii, ll, tt], q[ ii, ll, ll]}}, {{ q[ ff, ii, ii], q[ ff, ii, ff], q[ ff, ii, tt], q[ ff, ii, ll]}, { q[ ff, ff, ii], q[ ff, ff, ff], q[ ff, ff, tt], q[ ff, ff, ll]}, { q[ ff, tt, ii], q[ ff, tt, ff], q[ ff, tt, tt], q[ ff, tt, ll]}, { q[ ff, ll, ii], q[ ff, ll, ff], q[ ff, ll, tt], q[ ff, ll, ll]}}, {{ q[ tt, ii, ii], q[ tt, ii, ff], q[ tt, ii, tt], q[ tt, ii, ll]}, { q[ tt, ff, ii], q[ tt, ff, ff], q[ tt, ff, tt], q[ tt, ff, ll]}, { q[ tt, tt, ii], q[ tt, tt, ff], q[ tt, tt, tt], q[ tt, tt, ll]}, { q[ tt, ll, ii], q[ tt, ll, ff], q[ tt, ll, tt], q[ tt, ll, ll]}}, {{ q[ ll, ii, ii], q[ ll, ii, ff], q[ ll, ii, tt], q[ ll, ii, ll]}, { q[ ll, ff, ii], q[ ll, ff, ff], q[ ll, ff, tt], q[ ll, ff, ll]}, { q[ ll, tt, ii], q[ ll, tt, ff], q[ ll, tt, tt], q[ ll, tt, ll]}, { q[ ll, ll, ii], q[ ll, ll, ff], q[ ll, ll, tt], q[ ll, ll, ll]}}}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Dimensions[tag]\)], "Input"], Cell[BoxData[ \({4, 4, 4}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Transpose[{tag, idx}, {4, 1, 2, 3}] // TableForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {\(q[ii, ii, ii]\), "24"}, {\(q[ii, ii, ff]\), "19"}, {\(q[ii, ii, tt]\), "9"}, {\(q[ii, ii, ll]\), "1"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, ff, ii]\), "19"}, {\(q[ii, ff, ff]\), "24"}, {\(q[ii, ff, tt]\), "22"}, {\(q[ii, ff, ll]\), "12"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, tt, ii]\), "9"}, {\(q[ii, tt, ff]\), "21"}, {\(q[ii, tt, tt]\), "11"}, {\(q[ii, tt, ll]\), "4"} }, RowSpacings->0.25, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(q[ii, ll, ii]\), "1"}, 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