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Please wait.\ \>", "Print"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Quadratic terms in the coordinates ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " of the Riemannian curvature tensor: ", Cell[BoxData[ \(TraditionalForm\`R\_\(a\ b\ c\ d\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]] }], "Subtitle"], Cell[TextData[{ "The symmetry of the Riemannian curvature tensor is characterized by the \ Young symmetrizer of the standard tableau ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"1", "3"}, {"2", "4"} }], ")"}], TraditionalForm]]], ", which belongs to the partition \[Lambda] = [2,2].\nThen the quadratic \ terms are described by a left ideal L of \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_8\)]], "] whose structure can be determined from the Plethysm [2,2] \[CircleDot] \ [2]. Upper bounds of the multiplicities of this left ideal can be found by \ means of the Littlewood-Richardson product [2,2] [2,2]. We set:" }], "Text"], Cell[CellGroupData[{ Cell["riemsym = Parti[2,2]", "Input"], Cell[OutputFormData["\<\ Parti[2, 2]\ \>", "\<\ {2, 2}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["zwei = Parti[2]", "Input"], Cell[OutputFormData["\<\ Parti[2]\ \>", "\<\ {2}\ \>"], "Output"] }, Open ]], Cell["Now we have", "Text"], Cell[CellGroupData[{ Cell["lr22m22 = RLRule[riemsym,riemsym]", "Input"], Cell[OutputFormData["\<\ Parti[4, 4] + Parti[4, 2, 2] + Parti[4, 3, 1] + Parti[2, 2, 2, 2] + Parti[3, \ 2, 2, 1] + Parti[3, 3, 1, 1]\ \>", "\<\ {4, 4} + {4, 2, 2} + {4, 3, 1} + {2, 2, 2, 2} + {3, 2, 2, 1} + {3, 3, 1, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pleth22m2 = Plethysm[riemsym,zwei]", "Input"], Cell[OutputFormData["\<\ Parti[4, 4] + Parti[4, 2, 2] + Parti[2, 2, 2, 2] + Parti[3, 3, 1, 1]\ \>", "\<\ {4, 4} + {4, 2, 2} + {2, 2, 2, 2} + {3, 3, 1, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The number of minimal left ideals which occur in a decomposition of the \ full ring \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_8\)]], "] is 764." }], "Text"], Cell[CellGroupData[{ Cell["part8 = AllPartitions[8]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1], Parti[2, 2, 2, 1, 1], Parti[2, 2, 2, 2], Parti[3, 1, 1, 1, 1, 1], Parti[3, 2, 1, 1, 1], Parti[3, 2, 2, 1], Parti[3, \ 3, 1, 1], Parti[3, 3, 2], Parti[4, 1, 1, 1, 1], Parti[4, 2, 1, 1], Parti[4, 2, 2], Parti[4, 3, 1], Parti[4, 4], Parti[5, 1, 1, 1], Parti[5, 2, 1], Parti[5, \ 3], Parti[6, 1, 1], Parti[6, 2], Parti[7, 1], Parti[8]]\ \>", "\<\ {{1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1}, {2, 2, \ 2, 1, 1}, {2, 2, 2, 2}, {3, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1}, {3, 2, 2, 1}, {3, 3, 1, \ 1}, {3, 3, 2}, {4, 1, 1, 1, 1}, {4, 2, 1, 1}, {4, 2, 2}, {4, 3, 1}, {4, 4}, {5, \ 1, 1, 1}, {5, 2, 1}, {5, 3}, {6, 1, 1}, {6, 2}, {7, 1}, {8}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ part8", "Input"], Cell[OutputFormData["\<\ HoldList[1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, \ 28, 21, 20, 7, 1]\ \>", "\<\ {1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, 28, 21, \ 20, 7, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["allideals = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 764\ \>", "\<\ 764\ \>"], "Output"] }, Open ]], Cell["\<\ We determine the number of such ideals lying in equivalence classes of \ minimal left ideals which do not occur in lr22m22 or pleth22m2.\ \>", "Text"], Cell["Ad lr22m22:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ lr22m22", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"list\" is similar to existing symbol \"List\".\ \>", "Message"], Cell[OutputFormData["\<\ HoldList[Parti[4, 4], Parti[4, 2, 2], Parti[4, 3, 1], Parti[2, 2, 2, 2], Parti[3, 2, 2, 1], Parti[3, 3, 1, 1]]\ \>", "\<\ {{4, 4}, {4, 2, 2}, {4, 3, 1}, {2, 2, 2, 2}, {3, 2, 2, 1}, {3, 3, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part8,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8], Parti[5, 3], Parti[6, 2], Parti[7, 1], Parti[3, 3, 2], Parti[5, 2, 1], Parti[6, 1, 1], Parti[4, 2, 1, 1], Parti[5, 1, 1, 1], Parti[2, 2, 2, 1, 1], Parti[3, 2, 1, 1, 1], Parti[4, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1], Parti[3, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, \ 1], Parti[1, 1, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8}, {5, 3}, {6, 2}, {7, 1}, {3, 3, 2}, {5, 2, 1}, {6, 1, 1}, {4, 2, 1, 1}, {5, 1, 1, 1}, {2, 2, 2, 1, 1}, {3, 2, 1, 1, 1}, {4, 1, 1, 1, 1}, {2, 2, 1, \ 1, 1, 1}, {3, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 28, 20, 7, 42, 64, 21, 90, 35, 28, 64, 35, 20, 21, 7, 1]\ \>", "\<\ {1, 28, 20, 7, 42, 64, 21, 90, 35, 28, 64, 35, 20, 21, 7, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["idealnumber = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 484\ \>", "\<\ 484\ \>"], "Output"] }, Open ]], Cell["Ad pleth22m2:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ pleth22m2", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[4, 4], Parti[4, 2, 2], Parti[2, 2, 2, 2], Parti[3, 3, 1, 1]]\ \>", "\<\ {{4, 4}, {4, 2, 2}, {2, 2, 2, 2}, {3, 3, 1, 1}}\ \>"], "Output"] }, Open ]], Cell["\<\ Dimensions of the minimal left ideals belonging to these partitions:\ \>", "Text"], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ list", "Input"], Cell[OutputFormData["\<\ HoldList[14, 56, 14, 56]\ \>", "\<\ {14, 56, 14, 56}\ \>"], "Output"] }, Open ]], Cell[TextData[ "We determine the smallest and the largest dimension of an minimal left ideal \ within the decomposition of [2,2] \[CircleDot] [2]."], "Text"], Cell[CellGroupData[{ Cell["Min @@ dims", "Input"], Cell[OutputFormData["\<\ 14\ \>", "\<\ 14\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Max @@ dims", "Input"], Cell[OutputFormData["\<\ 56\ \>", "\<\ 56\ \>"], "Output"] }, Open ]], Cell[TextData[ "Further we calculate the memory (in MByte) which a 56 \[Times] 56-matrix \ filled with 2-Byte-Integers needs."], "Text"], Cell[CellGroupData[{ Cell["%*%*2 Byte", "Input"], Cell[OutputFormData["\<\ 6272*Byte\ \>", "\<\ 6272 Byte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 kByte/Byte)", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"kByte\" is similar to existing symbol \"Byte\".\ \>", "Message"], Cell[OutputFormData["\<\ (49*kByte)/8\ \>", "\<\ 49 kByte -------- 8\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 MByte/kByte)", "Input"], Cell["\<\ General::spell1: Possible spelling error: new symbol name \"MByte\" is similar to existing symbol \"Byte\".\ \>", "Message"], Cell[OutputFormData["\<\ (49*MByte)/8192\ \>", "\<\ 49 MByte -------- 8192\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["N[%]", "Input"], Cell[OutputFormData["\<\ 0.0059814453125*MByte\ \>", "\<\ 0.00598145 MByte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part8,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8], Parti[5, 3], Parti[6, 2], Parti[7, 1], Parti[3, 3, 2], Parti[4, 3, 1], Parti[5, 2, 1], Parti[6, 1, 1], Parti[3, 2, 2, 1], Parti[4, \ 2, 1, 1], Parti[5, 1, 1, 1], Parti[2, 2, 2, 1, 1], Parti[3, 2, 1, 1, 1], Parti[4, 1, \ 1, 1, 1], Parti[2, 2, 1, 1, 1, 1], Parti[3, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, \ 1], Parti[1, 1, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8}, {5, 3}, {6, 2}, {7, 1}, {3, 3, 2}, {4, 3, 1}, {5, 2, 1}, {6, 1, 1}, {3, \ 2, 2, 1}, {4, 2, 1, 1}, {5, 1, 1, 1}, {2, 2, 2, 1, 1}, {3, 2, 1, 1, 1}, {4, 1, 1, 1, \ 1}, {2, 2, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 28, 20, 7, 42, 70, 64, 21, 70, 90, 35, 28, 64, 35, 20, 21, 7, \ 1]\ \>", "\<\ {1, 28, 20, 7, 42, 70, 64, 21, 70, 90, 35, 28, 64, 35, 20, 21, 7, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[ "Now the number of minimal left ideals lying in equivalence classes which do \ not belong to [2,2] \[CircleDot] [2] is"], "Text"], Cell[CellGroupData[{ Cell["idealnumber = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 624\ \>", "\<\ 624\ \>"], "Output"] }, Open ]], Cell["\<\ An upper bound of the minimal left ideals within the decomposition of L is \ given by the minimal left ideals belonging to lr22m22.\ \>", "Text"], Cell[CellGroupData[{ Cell["idealnumber = lr22m22 /. Parti[___] -> 1", "Input"], Cell[OutputFormData["\<\ 6\ \>", "\<\ 6\ \>"], "Output"] }, Open ]], Cell["\<\ The real number of the minimal left ideals within the decomposition of L is \ given by the minimal left ideals belonging to pleth22m2.\ \>", "Text"], Cell[CellGroupData[{ Cell["idealnumber = pleth22m2 /. Parti[___] -> 1", "Input"], Cell[OutputFormData["\<\ 4\ \>", "\<\ 4\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Cubic terms in the coordinates ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " of the Riemannian curvature tensor: ", Cell[BoxData[ \(TraditionalForm\`R\_\(a\ b\ c\ d\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(p\ q\ r\ s\)\)]] }], "Subtitle"], Cell[TextData[{ "The cubic terms are described by a left ideal L of \ \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_12\)]], "] whose structure can be determined from the Plethysm [2,2] \[CircleDot] \ [3]. Upper bounds of the multiplicities of this left ideal can be found by \ means of the Littlewood-Richardson product [2,2] [2,2] [2,2]. We set:" }], "Text"], Cell[CellGroupData[{ Cell["drei = Parti[3]", "Input"], Cell[OutputFormData["\<\ Parti[3]\ \>", "\<\ {3}\ \>"], "Output"] }, Open ]], Cell["Now we have", "Text"], Cell[CellGroupData[{ Cell["lr22m22m22 = RLRule[lr22m22,riemsym]", "Input"], Cell[OutputFormData["\<\ Parti[6, 6] + Parti[4, 4, 4] + 2*Parti[5, 4, 3] + Parti[5, 5, 2] + Parti[6, \ 3, 3] + 3*Parti[6, 4, 2] + 2*Parti[6, 5, 1] + Parti[3, 3, 3, 3] + 3*Parti[4, 3, 3, \ 2] + 6*Parti[4, 4, 2, 2] + 3*Parti[4, 4, 3, 1] + 3*Parti[5, 3, 2, 2] + 3*Parti[5, 3, 3, 1] + 6*Parti[5, 4, 2, 1] + 3*Parti[5, 5, 1, 1] + Parti[6, \ 2, 2, 2] + 2*Parti[6, 3, 2, 1] + Parti[6, 4, 1, 1] + Parti[3, 3, 2, 2, 2] + 2*Parti[3, 3, 3, 2, 1] + 3*Parti[4, 2, 2, 2, 2] + 6*Parti[4, 3, 2, 2, 1] + 3*Parti[4, 3, 3, 1, 1] + 3*Parti[4, 4, 2, 1, 1] + 2*Parti[5, 2, 2, 2, 1] + 4*Parti[5, 3, 2, 1, 1] + 2*Parti[5, 4, 1, 1, 1] + Parti[2, 2, 2, 2, 2, 2] + \ 2*Parti[3, 2, 2, 2, 2, 1] + 3*Parti[3, 3, 2, 2, 1, 1] + Parti[3, 3, 3, 1, \ 1, 1] + Parti[4, 2, 2, 2, 1, 1] + 2*Parti[4, 3, 2, 1, 1, 1] + Parti[4, 4, 1, 1, 1, \ 1]\ \>", "\<\ {6, 6} + {4, 4, 4} + 2 {5, 4, 3} + {5, 5, 2} + {6, 3, 3} + 3 {6, 4, 2} + 2 \ {6, 5, 1} + {3, 3, 3, 3} + 3 {4, 3, 3, 2} + 6 {4, 4, 2, 2} + 3 {4, 4, 3, 1} + 3 {5, 3, \ 2, 2} + 3 {5, 3, 3, 1} + 6 {5, 4, 2, 1} + 3 {5, 5, 1, 1} + {6, 2, 2, 2} + 2 {6, 3, \ 2, 1} + {6, 4, 1, 1} + {3, 3, 2, 2, 2} + 2 {3, 3, 3, 2, 1} + 3 {4, 2, 2, 2, 2} + 6 {4, 3, 2, 2, 1} + 3 {4, 3, 3, 1, 1} + 3 {4, 4, 2, 1, 1} + 2 {5, 2, 2, 2, \ 1} + 4 {5, 3, 2, 1, 1} + 2 {5, 4, 1, 1, 1} + {2, 2, 2, 2, 2, 2} + 2 {3, 2, 2, 2, \ 2, 1} + 3 {3, 3, 2, 2, 1, 1} + {3, 3, 3, 1, 1, 1} + {4, 2, 2, 2, 1, 1} + 2 {4, 3, 2, 1, 1, 1} + {4, 4, 1, 1, 1, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pleth22m3 = Plethysm[riemsym,drei]", "Input"], Cell[OutputFormData["\<\ Parti[6, 6] + Parti[4, 4, 4] + Parti[6, 4, 2] + Parti[3, 3, 3, 3] + 2*Parti[4, 4, 2, 2] + Parti[5, 3, 3, 1] + Parti[5, 4, 2, 1] + Parti[5, 5, \ 1, 1] + Parti[6, 2, 2, 2] + Parti[4, 2, 2, 2, 2] + Parti[4, 3, 2, 2, 1] + Parti[4, 3, 3, 1, 1] + Parti[5, 3, 2, 1, 1] + Parti[2, 2, 2, 2, 2, 2] + Parti[3, 3, 2, 2, 1, 1] + Parti[4, 4, 1, 1, 1, 1]\ \>", "\<\ {6, 6} + {4, 4, 4} + {6, 4, 2} + {3, 3, 3, 3} + 2 {4, 4, 2, 2} + {5, 3, 3, 1} \ + {5, 4, 2, 1} + {5, 5, 1, 1} + {6, 2, 2, 2} + {4, 2, 2, 2, 2} + {4, 3, 2, 2, \ 1} + {4, 3, 3, 1, 1} + {5, 3, 2, 1, 1} + {2, 2, 2, 2, 2, 2} + {3, 3, 2, 2, 1, 1} \ + {4, 4, 1, 1, 1, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The number of minimal left ideals which occur in a decomposition of the \ full ring \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_12\)]], "] is 140152." }], "Text"], Cell[CellGroupData[{ Cell["part12 = AllPartitions[12]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1, 1, 1, 1, \ 1], Parti[2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 1, 1], Parti[2, 2, 2, 2, 2, 2], Parti[3, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 2, 1, 1, 1, 1, 1, 1, 1], Parti[3, 2, 2, 1, 1, 1, 1, 1], Parti[3, 2, 2, 2, 1, 1, 1], Parti[3, 2, 2, \ 2, 2, 1], Parti[3, 3, 1, 1, 1, 1, 1, 1], Parti[3, 3, 2, 1, 1, 1, 1], Parti[3, 3, 2, \ 2, 1, 1], Parti[3, 3, 2, 2, 2], Parti[3, 3, 3, 1, 1, 1], Parti[3, 3, 3, 2, 1], Parti[3, 3, 3, 3], Parti[4, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 2, 1, 1, 1, \ 1, 1, 1], Parti[4, 2, 2, 1, 1, 1, 1], Parti[4, 2, 2, 2, 1, 1], Parti[4, 2, 2, 2, 2], Parti[4, 3, 1, 1, 1, 1, 1], Parti[4, 3, 2, 1, 1, 1], Parti[4, 3, 2, 2, 1], Parti[4, 3, 3, 1, 1], Parti[4, 3, 3, 2], Parti[4, 4, 1, 1, 1, 1], Parti[4, 4, 2, 1, 1], Parti[4, 4, 2, 2], Parti[4, 4, 3, 1], Parti[4, 4, 4], \ Parti[5, 1, 1, 1, 1, 1, 1, 1], Parti[5, 2, 1, 1, 1, 1, 1], Parti[5, 2, 2, \ 1, 1, 1], Parti[5, 2, 2, 2, 1], Parti[5, 3, 1, 1, 1, 1], Parti[5, 3, 2, 1, 1], Parti[5, 3, 2, 2], Parti[5, 3, 3, 1], Parti[5, 4, 1, 1, 1], Parti[5, 4, 2, \ 1], Parti[5, 4, 3], Parti[5, 5, 1, 1], Parti[5, 5, 2], Parti[6, 1, 1, 1, 1, 1, \ 1], Parti[6, 2, 1, 1, 1, 1], Parti[6, 2, 2, 1, 1], Parti[6, 2, 2, 2], Parti[6, 3, 1, 1, 1], Parti[6, 3, 2, 1], Parti[6, 3, 3], Parti[6, 4, 1, 1], \ Parti[6, 4, 2], Parti[6, 5, 1], Parti[6, 6], Parti[7, 1, 1, 1, 1, 1], Parti[7, 2, 1, 1, 1], Parti[7, 2, 2, 1], Parti[7, 3, 1, 1], Parti[7, 3, 2], \ Parti[7, 4, 1], Parti[7, 5], Parti[8, 1, 1, 1, 1], Parti[8, 2, 1, 1], \ Parti[8, 2, 2], Parti[8, 3, 1], Parti[8, 4], Parti[9, 1, 1, 1], Parti[9, 2, 1], Parti[9, \ 3], Parti[10, 1, 1], Parti[10, 2], Parti[11, 1], Parti[12]]\ \>", "\<\ {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, \ 1, 1, 1, 1}, {2, 2, 2, 2, 2, 1, 1}, {2, 2, 2, 2, 2, 2}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1}, {3, 2, 2, 2, 1, 1, \ 1}, {3, 2, 2, 2, 2, 1}, {3, 3, 1, 1, 1, 1, 1, 1}, {3, 3, 2, 1, 1, 1, 1}, {3, 3, 2, 2, 1, 1}, {3, 3, 2, 2, 2}, {3, 3, 3, 1, 1, 1}, {3, 3, 3, 2, 1}, {3, 3, 3, 3}, {4, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 1, 1, 1, 1, 1, 1}, {4, 2, 2, 1, 1, 1, 1}, {4, 2, 2, 2, 1, 1}, {4, 2, 2, 2, 2}, {4, 3, 1, 1, 1, \ 1, 1}, {4, 3, 2, 1, 1, 1}, {4, 3, 2, 2, 1}, {4, 3, 3, 1, 1}, {4, 3, 3, 2}, {4, 4, 1, 1, 1, 1}, {4, 4, 2, 1, 1}, {4, 4, 2, 2}, {4, 4, 3, 1}, {4, 4, 4}, \ {5, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1}, {5, 2, 2, 1, 1, 1}, {5, 2, \ 2, 2, 1}, {5, 3, 1, 1, 1, 1}, {5, 3, 2, 1, 1}, {5, 3, 2, 2}, {5, 3, 3, 1}, {5, 4, 1, \ 1, 1}, {5, 4, 2, 1}, {5, 4, 3}, {5, 5, 1, 1}, {5, 5, 2}, {6, 1, 1, 1, 1, 1, 1}, {6, 2, 1, 1, 1, 1}, {6, 2, 2, 1, 1}, {6, 2, 2, 2}, {6, 3, 1, 1, 1}, {6, 3, \ 2, 1}, {6, 3, 3}, {6, 4, 1, 1}, {6, 4, 2}, {6, 5, 1}, {6, 6}, {7, 1, 1, 1, 1, 1}, {7, 2, 1, 1, 1}, {7, 2, 2, 1}, {7, 3, 1, 1}, {7, 3, 2}, {7, 4, 1}, {7, 5}, {8, 1, 1, 1, 1}, {8, 2, 1, 1}, {8, 2, 2}, {8, 3, 1}, {8, 4}, {9, 1, 1, 1}, \ {9, 2, 1}, {9, 3}, {10, 1, 1}, {10, 2}, {11, 1}, {12}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ part12", "Input"], Cell[OutputFormData["\<\ HoldList[1, 11, 54, 154, 275, 297, 132, 55, 320, 891, 1408, 1155, 616, 1925, \ 2673, 1320, 1650, 2112, 462, 165, 945, 2376, 3080, 1485, 2079, 5632, 5775, 4158, 2970, \ 1925, 4455, 2640, 2970, 462, 330, 1728, 3696, 3520, 3564, 7700, 4455, 4158, 3520, 5775, \ 2112, 1485, 1320, 462, 2100, 3564, 1925, 3696, 5632, 1650, 3080, 2673, 1155, 132, \ 462, 1728, 2079, 2376, 1925, 1408, 297, 330, 945, 616, 891, 275, 165, 320, 154, 55, \ 54, 11, 1]\ \>", "\<\ {1, 11, 54, 154, 275, 297, 132, 55, 320, 891, 1408, 1155, 616, 1925, 2673, \ 1320, 1650, 2112, 462, 165, 945, 2376, 3080, 1485, 2079, 5632, 5775, 4158, 2970, 1925, \ 4455, 2640, 2970, 462, 330, 1728, 3696, 3520, 3564, 7700, 4455, 4158, 3520, 5775, 2112, \ 1485, 1320, 462, 2100, 3564, 1925, 3696, 5632, 1650, 3080, 2673, 1155, 132, 462, \ 1728, 2079, 2376, 1925, 1408, 297, 330, 945, 616, 891, 275, 165, 320, 154, 55, 54, 11, \ 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["allideals = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 140152\ \>", "\<\ 140152\ \>"], "Output"] }, Open ]], Cell["\<\ We determine the number of such ideals lying in equivalence classes of \ minimal left ideals which do not occur in lr22m22m22 or pleth22m3.\ \>", "Text"], Cell["Ad lr22m22m22:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ lr22m22m22", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[6, 6], Parti[4, 4, 4], 2*Parti[5, 4, 3], Parti[5, 5, 2], \ Parti[6, 3, 3], 3*Parti[6, 4, 2], 2*Parti[6, 5, 1], Parti[3, 3, 3, 3], 3*Parti[4, 3, 3, 2], \ 6*Parti[4, 4, 2, 2], 3*Parti[4, 4, 3, 1], 3*Parti[5, 3, 2, 2], 3*Parti[5, \ 3, 3, 1], 6*Parti[5, 4, 2, 1], 3*Parti[5, 5, 1, 1], Parti[6, 2, 2, 2], 2*Parti[6, 3, \ 2, 1], Parti[6, 4, 1, 1], Parti[3, 3, 2, 2, 2], 2*Parti[3, 3, 3, 2, 1], 3*Parti[4, 2, 2, 2, 2], 6*Parti[4, 3, 2, 2, 1], 3*Parti[4, 3, 3, 1, 1], 3*Parti[4, 4, 2, 1, 1], 2*Parti[5, 2, 2, 2, 1], 4*Parti[5, 3, 2, 1, 1], 2*Parti[5, 4, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2], 2*Parti[3, 2, 2, 2, 2, 1], \ 3*Parti[3, 3, 2, 2, 1, 1], Parti[3, 3, 3, 1, 1, 1], Parti[4, 2, 2, 2, 1, \ 1], 2*Parti[4, 3, 2, 1, 1, 1], Parti[4, 4, 1, 1, 1, 1]]\ \>", "\<\ {{6, 6}, {4, 4, 4}, 2 {5, 4, 3}, {5, 5, 2}, {6, 3, 3}, 3 {6, 4, 2}, 2 {6, 5, \ 1}, {3, 3, 3, 3}, 3 {4, 3, 3, 2}, 6 {4, 4, 2, 2}, 3 {4, 4, 3, 1}, 3 {5, 3, 2, \ 2}, 3 {5, 3, 3, 1}, 6 {5, 4, 2, 1}, 3 {5, 5, 1, 1}, {6, 2, 2, 2}, 2 {6, 3, 2, \ 1}, {6, 4, 1, 1}, {3, 3, 2, 2, 2}, 2 {3, 3, 3, 2, 1}, 3 {4, 2, 2, 2, 2}, 6 {4, 3, 2, 2, 1}, 3 {4, 3, 3, 1, 1}, 3 {4, 4, 2, 1, 1}, 2 {5, 2, 2, 2, 1}, \ 4 {5, 3, 2, 1, 1}, 2 {5, 4, 1, 1, 1}, {2, 2, 2, 2, 2, 2}, 2 {3, 2, 2, 2, 2, \ 1}, 3 {3, 3, 2, 2, 1, 1}, {3, 3, 3, 1, 1, 1}, {4, 2, 2, 2, 1, 1}, 2 {4, 3, 2, \ 1, 1, 1}, {4, 4, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 34\ \>", "\<\ 34\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["list = list /. _*x_Parti :> x", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[6, 6], Parti[4, 4, 4], Parti[5, 4, 3], Parti[5, 5, 2], \ Parti[6, 3, 3], Parti[6, 4, 2], Parti[6, 5, 1], Parti[3, 3, 3, 3], Parti[4, 3, 3, 2], Parti[4, 4, 2, 2], Parti[4, 4, 3, 1], Parti[5, 3, 2, 2], Parti[5, 3, 3, 1], \ Parti[5, 4, 2, 1], Parti[5, 5, 1, 1], Parti[6, 2, 2, 2], Parti[6, 3, 2, 1], \ Parti[6, 4, 1, 1], Parti[3, 3, 2, 2, 2], Parti[3, 3, 3, 2, 1], Parti[4, 2, \ 2, 2, 2], Parti[4, 3, 2, 2, 1], Parti[4, 3, 3, 1, 1], Parti[4, 4, 2, 1, 1], Parti[5, 2, 2, 2, 1], Parti[5, 3, 2, 1, 1], Parti[5, 4, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2], Parti[3, 2, 2, 2, 2, 1], Parti[3, 3, 2, 2, 1, 1], Parti[3, 3, 3, 1, 1, 1], Parti[4, 2, 2, 2, 1, 1], Parti[4, 3, 2, 1, 1, 1], Parti[4, 4, 1, 1, 1, 1]]\ \>", "\<\ {{6, 6}, {4, 4, 4}, {5, 4, 3}, {5, 5, 2}, {6, 3, 3}, {6, 4, 2}, {6, 5, 1}, \ {3, 3, 3, 3}, {4, 3, 3, 2}, {4, 4, 2, 2}, {4, 4, 3, 1}, {5, 3, 2, 2}, {5, 3, 3, 1}, {5, \ 4, 2, 1}, {5, 5, 1, 1}, {6, 2, 2, 2}, {6, 3, 2, 1}, {6, 4, 1, 1}, {3, 3, 2, 2, 2}, {3, 3, 3, 2, 1}, {4, 2, 2, 2, 2}, {4, 3, 2, 2, 1}, {4, 3, 3, 1, 1}, {4, 4, \ 2, 1, 1}, {5, 2, 2, 2, 1}, {5, 3, 2, 1, 1}, {5, 4, 1, 1, 1}, {2, 2, 2, 2, 2, 2}, {3, 2, 2, 2, 2, 1}, {3, 3, 2, 2, 1, 1}, {3, 3, 3, 1, 1, 1}, {4, 2, 2, 2, 1, \ 1}, {4, 3, 2, 1, 1, 1}, {4, 4, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 34\ \>", "\<\ 34\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part12,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[12], Parti[7, 5], Parti[8, 4], Parti[9, 3], Parti[10, 2], \ Parti[11, 1], Parti[7, 3, 2], Parti[7, 4, 1], Parti[8, 2, 2], Parti[8, 3, 1], Parti[9, 2, \ 1], Parti[10, 1, 1], Parti[7, 2, 2, 1], Parti[7, 3, 1, 1], Parti[8, 2, 1, 1], Parti[9, 1, 1, 1], Parti[6, 2, 2, 1, 1], Parti[6, 3, 1, 1, 1], Parti[7, 2, \ 1, 1, 1], Parti[8, 1, 1, 1, 1], Parti[5, 2, 2, 1, 1, 1], Parti[5, 3, 1, 1, 1, 1], Parti[6, 2, 1, 1, 1, 1], Parti[7, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 1, \ 1], Parti[3, 2, 2, 2, 1, 1, 1], Parti[3, 3, 2, 1, 1, 1, 1], Parti[4, 2, 2, 1, \ 1, 1, 1], Parti[4, 3, 1, 1, 1, 1, 1], Parti[5, 2, 1, 1, 1, 1, 1], Parti[6, 1, 1, 1, \ 1, 1, 1], Parti[2, 2, 2, 2, 1, 1, 1, 1], Parti[3, 2, 2, 1, 1, 1, 1, 1], Parti[3, 3, 1, 1, 1, 1, 1, 1], Parti[4, 2, 1, 1, 1, 1, 1, 1], Parti[5, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[3, 2, 1, 1, 1, 1, 1, 1, 1], Parti[4, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1]]\ \>", "\<\ {{12}, {7, 5}, {8, 4}, {9, 3}, {10, 2}, {11, 1}, {7, 3, 2}, {7, 4, 1}, {8, 2, \ 2}, {8, 3, 1}, {9, 2, 1}, {10, 1, 1}, {7, 2, 2, 1}, {7, 3, 1, 1}, {8, 2, 1, 1}, \ {9, 1, 1, 1}, {6, 2, 2, 1, 1}, {6, 3, 1, 1, 1}, {7, 2, 1, 1, 1}, {8, 1, 1, \ 1, 1}, {5, 2, 2, 1, 1, 1}, {5, 3, 1, 1, 1, 1}, {6, 2, 1, 1, 1, 1}, {7, 1, 1, 1, 1, \ 1}, {2, 2, 2, 2, 2, 1, 1}, {3, 2, 2, 2, 1, 1, 1}, {3, 3, 2, 1, 1, 1, 1}, {4, 2, 2, 1, 1, 1, 1}, {4, 3, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1}, {6, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 1, 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1}, {3, 3, 1, 1, 1, 1, 1, 1}, {4, 2, 1, 1, 1, 1, 1, 1}, {5, 1, 1, 1, 1, 1, 1, \ 1}, {2, 2, 2, 1, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1, 1, 1, 1, 1}, {4, 1, 1, 1, 1, \ 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 297, 275, 154, 54, 11, 1925, 1408, 616, 891, 320, 55, 2079, 2376, \ 945, 165, 3564, 3696, 1728, 330, 3696, 3564, 2100, 462, 297, 1408, 1925, 2376, 2079, \ 1728, 462, 275, 891, 616, 945, 330, 154, 320, 165, 54, 55, 11, 1]\ \>", "\<\ {1, 297, 275, 154, 54, 11, 1925, 1408, 616, 891, 320, 55, 2079, 2376, 945, \ 165, 3564, 3696, 1728, 330, 3696, 3564, 2100, 462, 297, 1408, 1925, 2376, 2079, 1728, \ 462, 275, 891, 616, 945, 330, 154, 320, 165, 54, 55, 11, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["idealnumber = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 44804\ \>", "\<\ 44804\ \>"], "Output"] }, Open ]], Cell["Ad pleth22m3:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ pleth22m3", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[6, 6], Parti[4, 4, 4], Parti[6, 4, 2], Parti[3, 3, 3, 3], 2*Parti[4, 4, 2, 2], Parti[5, 3, 3, 1], Parti[5, 4, 2, 1], Parti[5, 5, 1, \ 1], Parti[6, 2, 2, 2], Parti[4, 2, 2, 2, 2], Parti[4, 3, 2, 2, 1], Parti[4, 3, \ 3, 1, 1], Parti[5, 3, 2, 1, 1], Parti[2, 2, 2, 2, 2, 2], Parti[3, 3, 2, 2, 1, 1], Parti[4, 4, 1, 1, 1, 1]]\ \>", "\<\ {{6, 6}, {4, 4, 4}, {6, 4, 2}, {3, 3, 3, 3}, 2 {4, 4, 2, 2}, {5, 3, 3, 1}, \ {5, 4, 2, 1}, {5, 5, 1, 1}, {6, 2, 2, 2}, {4, 2, 2, 2, 2}, {4, 3, 2, 2, 1}, {4, 3, 3, 1, \ 1}, {5, 3, 2, 1, 1}, {2, 2, 2, 2, 2, 2}, {3, 3, 2, 2, 1, 1}, {4, 4, 1, 1, 1, \ 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 16\ \>", "\<\ 16\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["list = list /. _*x_Parti :> x", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[6, 6], Parti[4, 4, 4], Parti[6, 4, 2], Parti[3, 3, 3, 3], Parti[4, 4, 2, 2], Parti[5, 3, 3, 1], Parti[5, 4, 2, 1], Parti[5, 5, 1, 1], \ Parti[6, 2, 2, 2], Parti[4, 2, 2, 2, 2], Parti[4, 3, 2, 2, 1], Parti[4, 3, \ 3, 1, 1], Parti[5, 3, 2, 1, 1], Parti[2, 2, 2, 2, 2, 2], Parti[3, 3, 2, 2, 1, 1], Parti[4, 4, 1, 1, 1, 1]]\ \>", "\<\ {{6, 6}, {4, 4, 4}, {6, 4, 2}, {3, 3, 3, 3}, {4, 4, 2, 2}, {5, 3, 3, 1}, {5, \ 4, 2, 1}, {5, 5, 1, 1}, {6, 2, 2, 2}, {4, 2, 2, 2, 2}, {4, 3, 2, 2, 1}, {4, 3, 3, 1, \ 1}, {5, 3, 2, 1, 1}, {2, 2, 2, 2, 2, 2}, {3, 3, 2, 2, 1, 1}, {4, 4, 1, 1, 1, \ 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 16\ \>", "\<\ 16\ \>"], "Output"] }, Open ]], Cell["\<\ Dimensions of the minimal left ideals belonging to these partitions:\ \>", "Text"], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ list", "Input"], Cell[OutputFormData["\<\ HoldList[132, 462, 2673, 462, 2640, 4158, 5775, 1485, 1925, 1485, 5775, 4158, \ 7700, 132, 2673, 1925]\ \>", "\<\ {132, 462, 2673, 462, 2640, 4158, 5775, 1485, 1925, 1485, 5775, 4158, 7700, \ 132, 2673, 1925}\ \>"], "Output"] }, Open ]], Cell[TextData[ "We determine the smallest and the largest dimension of an minimal left ideal \ within the decomposition of [2,2] \[CircleDot] [3]."], "Text"], Cell[CellGroupData[{ Cell["Min @@ dims", "Input"], Cell[OutputFormData["\<\ 132\ \>", "\<\ 132\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Max @@ dims", "Input"], Cell[OutputFormData["\<\ 7700\ \>", "\<\ 7700\ \>"], "Output"] }, Open ]], Cell[TextData[ "Further we calculate the memory (in MByte) which a 7700 \[Times] 7700-matrix \ filled with 2-Byte-Integers needs."], "Text"], Cell[CellGroupData[{ Cell["%*%*2 Byte", "Input"], Cell[OutputFormData["\<\ 118580000*Byte\ \>", "\<\ 118580000 Byte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 kByte/Byte)", "Input"], Cell[OutputFormData["\<\ (3705625*kByte)/32\ \>", "\<\ 3705625 kByte ------------- 32\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 MByte/kByte)", "Input"], Cell[OutputFormData["\<\ (3705625*MByte)/32768\ \>", "\<\ 3705625 MByte ------------- 32768\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["N[%]", "Input"], Cell[OutputFormData["\<\ 113.0867004394531*MByte\ \>", "\<\ 113.087 MByte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part12,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[12], Parti[7, 5], Parti[8, 4], Parti[9, 3], Parti[10, 2], \ Parti[11, 1], Parti[5, 4, 3], Parti[5, 5, 2], Parti[6, 3, 3], Parti[6, 5, 1], Parti[7, 3, \ 2], Parti[7, 4, 1], Parti[8, 2, 2], Parti[8, 3, 1], Parti[9, 2, 1], Parti[10, \ 1, 1], Parti[4, 3, 3, 2], Parti[4, 4, 3, 1], Parti[5, 3, 2, 2], Parti[6, 3, 2, 1], \ Parti[6, 4, 1, 1], Parti[7, 2, 2, 1], Parti[7, 3, 1, 1], Parti[8, 2, 1, 1], \ Parti[9, 1, 1, 1], Parti[3, 3, 2, 2, 2], Parti[3, 3, 3, 2, 1], Parti[4, 4, \ 2, 1, 1], Parti[5, 2, 2, 2, 1], Parti[5, 4, 1, 1, 1], Parti[6, 2, 2, 1, 1], Parti[6, 3, 1, 1, 1], Parti[7, 2, 1, 1, 1], Parti[8, 1, 1, 1, 1], Parti[3, 2, 2, 2, 2, 1], Parti[3, 3, 3, 1, 1, 1], Parti[4, 2, 2, 2, 1, 1], Parti[4, 3, 2, 1, 1, 1], Parti[5, 2, 2, 1, 1, 1], Parti[5, 3, 1, 1, 1, 1], Parti[6, 2, 1, 1, 1, 1], Parti[7, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 1, \ 1], Parti[3, 2, 2, 2, 1, 1, 1], Parti[3, 3, 2, 1, 1, 1, 1], Parti[4, 2, 2, 1, \ 1, 1, 1], Parti[4, 3, 1, 1, 1, 1, 1], Parti[5, 2, 1, 1, 1, 1, 1], Parti[6, 1, 1, 1, \ 1, 1, 1], Parti[2, 2, 2, 2, 1, 1, 1, 1], Parti[3, 2, 2, 1, 1, 1, 1, 1], Parti[3, 3, 1, 1, 1, 1, 1, 1], Parti[4, 2, 1, 1, 1, 1, 1, 1], Parti[5, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[3, 2, 1, 1, 1, 1, 1, 1, 1], Parti[4, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1]]\ \>", "\<\ {{12}, {7, 5}, {8, 4}, {9, 3}, {10, 2}, {11, 1}, {5, 4, 3}, {5, 5, 2}, {6, 3, \ 3}, {6, 5, 1}, {7, 3, 2}, {7, 4, 1}, {8, 2, 2}, {8, 3, 1}, {9, 2, 1}, {10, 1, \ 1}, {4, 3, 3, 2}, {4, 4, 3, 1}, {5, 3, 2, 2}, {6, 3, 2, 1}, {6, 4, 1, 1}, {7, \ 2, 2, 1}, {7, 3, 1, 1}, {8, 2, 1, 1}, {9, 1, 1, 1}, {3, 3, 2, 2, 2}, {3, 3, 3, 2, 1}, \ {4, 4, 2, 1, 1}, {5, 2, 2, 2, 1}, {5, 4, 1, 1, 1}, {6, 2, 2, 1, 1}, {6, 3, \ 1, 1, 1}, {7, 2, 1, 1, 1}, {8, 1, 1, 1, 1}, {3, 2, 2, 2, 2, 1}, {3, 3, 3, 1, 1, 1}, {4, 2, 2, 2, 1, 1}, {4, 3, 2, 1, 1, 1}, {5, 2, 2, 1, 1, 1}, {5, 3, 1, 1, 1, \ 1}, {6, 2, 1, 1, 1, 1}, {7, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 1, 1}, {3, 2, 2, 2, \ 1, 1, 1}, {3, 3, 2, 1, 1, 1, 1}, {4, 2, 2, 1, 1, 1, 1}, {4, 3, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1}, {6, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 1, 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1}, {3, 3, 1, 1, 1, 1, 1, 1}, {4, 2, 1, 1, 1, 1, 1, \ 1}, {5, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 1, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1, 1, \ 1, 1, 1}, {4, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 297, 275, 154, 54, 11, 2112, 1320, 1650, 1155, 1925, 1408, 616, \ 891, 320, 55, 2970, 2970, 4455, 5632, 3080, 2079, 2376, 945, 165, 1320, 2112, 4455, \ 3520, 3520, 3564, 3696, 1728, 330, 1155, 1650, 3080, 5632, 3696, 3564, 2100, 462, 297, \ 1408, 1925, 2376, 2079, 1728, 462, 275, 891, 616, 945, 330, 154, 320, 165, 54, 55, 11, \ 1]\ \>", "\<\ {1, 297, 275, 154, 54, 11, 2112, 1320, 1650, 1155, 1925, 1408, 616, 891, 320, \ 55, 2970, 2970, 4455, 5632, 3080, 2079, 2376, 945, 165, 1320, 2112, 4455, 3520, 3520, \ 3564, 3696, 1728, 330, 1155, 1650, 3080, 5632, 3696, 3564, 2100, 462, 297, 1408, \ 1925, 2376, 2079, 1728, 462, 275, 891, 616, 945, 330, 154, 320, 165, 54, 55, 11, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[ "Now the number of minimal left ideals lying in equivalence classes which do \ not belong to [2,2] \[CircleDot] [3] is"], "Text"], Cell[CellGroupData[{ Cell["idealnumber = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 96592\ \>", "\<\ 96592\ \>"], "Output"] }, Open ]], Cell["\<\ An upper bound of the minimal left ideals within the decomposition of L is \ given by the minimal left ideals belonging to lr22m22m22.\ \>", "Text"], Cell[CellGroupData[{ Cell["idealnumber = lr22m22m22 /. Parti[___] -> 1", "Input"], Cell[OutputFormData["\<\ 80\ \>", "\<\ 80\ \>"], "Output"] }, Open ]], Cell["\<\ The real number of the minimal left ideals within the decomposition of L is \ given by the minimal left ideals belonging to pleth22m3.\ \>", "Text"], Cell[CellGroupData[{ Cell["idealnumber = pleth22m3 /. Parti[___] -> 1", "Input"], Cell[OutputFormData["\<\ 17\ \>", "\<\ 17\ \>"], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "4th order terms in the coordinates ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " of the Riemannian curvature tensor: ", Cell[BoxData[ \(TraditionalForm\`R\_\(a\ b\ c\ d\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(i\ j\ k\ l\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(p\ q\ r\ s\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`R\_\(t\ u\ v\ w\)\)]] }], "Subtitle"], Cell[TextData[{ "The 4th order terms are described by a left ideal L of \ \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_16\)]], "] whose structure can be determined from the Plethysm [2,2] \[CircleDot] \ [4]. Upper bounds of the multiplicities of this left ideal can be found by \ means of the Littlewood-Richardson product [2,2] [2,2] [2,2] [2,2]. We set:" }], "Text"], Cell[CellGroupData[{ Cell["vier = Parti[4]", "Input"], Cell[OutputFormData["\<\ Parti[4]\ \>", "\<\ {4}\ \>"], "Output"] }, Open ]], Cell["Now we have", "Text"], Cell[CellGroupData[{ Cell["lr22m22m22m22 = RLRule[lr22m22m22,riemsym]", "Input"], Cell[OutputFormData["\<\ Parti[8, 8] + 3*Parti[6, 5, 5] + 6*Parti[6, 6, 4] + 7*Parti[7, 5, 4] + 8*Parti[7, 6, 3] + 3*Parti[7, 7, 2] + 3*Parti[8, 4, 4] + 6*Parti[8, 5, 3] + \ 6*Parti[8, 6, 2] + 3*Parti[8, 7, 1] + 6*Parti[4, 4, 4, 4] + 21*Parti[5, 4, \ 4, 3] + 22*Parti[5, 5, 3, 3] + 21*Parti[5, 5, 4, 2] + 5*Parti[5, 5, 5, 1] + 21*Parti[6, 4, 3, 3] + 30*Parti[6, 4, 4, 2] + 45*Parti[6, 5, 3, 2] + 24*Parti[6, 5, 4, 1] + 21*Parti[6, 6, 2, 2] + 18*Parti[6, 6, 3, 1] + 5*Parti[7, 3, 3, 3] + 24*Parti[7, 4, 3, 2] + 14*Parti[7, 4, 4, 1] + 18*Parti[7, 5, 2, 2] + 27*Parti[7, 5, 3, 1] + 19*Parti[7, 6, 2, 1] + 6*Parti[7, 7, 1, 1] + 3*Parti[8, 3, 3, 2] + 6*Parti[8, 4, 2, 2] + 7*Parti[8, 4, 3, 1] + 8*Parti[8, 5, 2, 1] + 3*Parti[8, 6, 1, 1] + 5*Parti[4, 3, 3, 3, 3] + 21*Parti[4, 4, 3, 3, 2] + 22*Parti[4, 4, 4, 2, 2] \ + 21*Parti[4, 4, 4, 3, 1] + 22*Parti[5, 3, 3, 3, 2] + 56*Parti[5, 4, 3, 2, 2] \ + 48*Parti[5, 4, 3, 3, 1] + 48*Parti[5, 4, 4, 2, 1] + 20*Parti[5, 5, 2, 2, 2] \ + 56*Parti[5, 5, 3, 2, 1] + 22*Parti[5, 5, 4, 1, 1] + 24*Parti[6, 3, 3, 2, 2] \ + 18*Parti[6, 3, 3, 3, 1] + 36*Parti[6, 4, 2, 2, 2] + 72*Parti[6, 4, 3, 2, 1] \ + 24*Parti[6, 4, 4, 1, 1] + 48*Parti[6, 5, 2, 2, 1] + 42*Parti[6, 5, 3, 1, 1] \ + 18*Parti[6, 6, 2, 1, 1] + 8*Parti[7, 3, 2, 2, 2] + 16*Parti[7, 3, 3, 2, 1] \ + 24*Parti[7, 4, 2, 2, 1] + 24*Parti[7, 4, 3, 1, 1] + 24*Parti[7, 5, 2, 1, 1] \ + 8*Parti[7, 6, 1, 1, 1] + Parti[8, 2, 2, 2, 2] + 3*Parti[8, 3, 2, 2, 1] + 2*Parti[8, 3, 3, 1, 1] + 3*Parti[8, 4, 2, 1, 1] + Parti[8, 5, 1, 1, 1] + 6*Parti[3, 3, 3, 3, 2, 2] + 3*Parti[3, 3, 3, 3, 3, 1] + 18*Parti[4, 3, 3, \ 2, 2, 2] + 24*Parti[4, 3, 3, 3, 2, 1] + 21*Parti[4, 4, 2, 2, 2, 2] + 45*Parti[4, 4, 3, \ 2, 2, 1] + 30*Parti[4, 4, 3, 3, 1, 1] + 21*Parti[4, 4, 4, 2, 1, 1] + 18*Parti[5, 3, 2, \ 2, 2, 2] + 42*Parti[5, 3, 3, 2, 2, 1] + 24*Parti[5, 3, 3, 3, 1, 1] + 48*Parti[5, 4, 2, \ 2, 2, 1] + 72*Parti[5, 4, 3, 2, 1, 1] + 18*Parti[5, 4, 4, 1, 1, 1] + 36*Parti[5, 5, 2, \ 2, 1, 1] + 24*Parti[5, 5, 3, 1, 1, 1] + 6*Parti[6, 2, 2, 2, 2, 2] + 24*Parti[6, 3, 2, \ 2, 2, 1] + 30*Parti[6, 3, 3, 2, 1, 1] + 36*Parti[6, 4, 2, 2, 1, 1] + 30*Parti[6, 4, 3, \ 1, 1, 1] + 24*Parti[6, 5, 2, 1, 1, 1] + 6*Parti[6, 6, 1, 1, 1, 1] + 3*Parti[7, 2, 2, \ 2, 2, 1] + 9*Parti[7, 3, 2, 2, 1, 1] + 6*Parti[7, 3, 3, 1, 1, 1] + 9*Parti[7, 4, 2, 1, \ 1, 1] + 3*Parti[7, 5, 1, 1, 1, 1] + 3*Parti[3, 3, 2, 2, 2, 2, 2] + 8*Parti[3, 3, 3, 2, 2, 2, 1] + 7*Parti[3, 3, 3, 3, 2, 1, 1] + 6*Parti[4, 2, 2, 2, 2, 2, 2] + 19*Parti[4, 3, 2, 2, 2, 2, 1] + 27*Parti[4, 3, 3, 2, 2, 1, 1] + 14*Parti[4, 3, 3, 3, 1, 1, 1] + 18*Parti[4, 4, 2, 2, 2, 1, 1] + 24*Parti[4, 4, 3, 2, 1, 1, 1] + 5*Parti[4, 4, 4, 1, 1, 1, 1] + 8*Parti[5, 2, 2, 2, 2, 2, 1] + 24*Parti[5, 3, 2, 2, 2, 1, 1] + 24*Parti[5, 3, 3, 2, 1, 1, 1] + 24*Parti[5, 4, 2, 2, 1, 1, 1] + 16*Parti[5, 4, 3, 1, 1, 1, 1] + 8*Parti[5, 5, 2, 1, 1, 1, 1] + 3*Parti[6, 2, 2, 2, 2, 1, 1] + 9*Parti[6, 3, 2, 2, 1, 1, 1] + 6*Parti[6, 3, 3, 1, 1, 1, 1] + 9*Parti[6, 4, 2, 1, 1, 1, 1] + 3*Parti[6, 5, 1, 1, 1, 1, 1] + Parti[2, 2, 2, 2, 2, 2, 2, 2] + 3*Parti[3, 2, 2, 2, 2, 2, 2, 1] + 6*Parti[3, 3, 2, 2, 2, 2, 1, 1] + 6*Parti[3, 3, 3, 2, 2, 1, 1, 1] + 3*Parti[3, 3, 3, 3, 1, 1, 1, 1] + 3*Parti[4, 2, 2, 2, 2, 2, 1, 1] + 8*Parti[4, 3, 2, 2, 2, 1, 1, 1] + 7*Parti[4, 3, 3, 2, 1, 1, 1, 1] + 6*Parti[4, 4, 2, 2, 1, 1, 1, 1] + 3*Parti[4, 4, 3, 1, 1, 1, 1, 1] + Parti[5, 2, 2, 2, 2, 1, 1, 1] + 3*Parti[5, 3, 2, 2, 1, 1, 1, 1] + 2*Parti[5, 3, 3, 1, 1, 1, 1, 1] + 3*Parti[5, 4, 2, 1, 1, 1, 1, 1] + Parti[5, 5, 1, 1, 1, 1, 1, 1]\ \>", "\<\ {8, 8} + 3 {6, 5, 5} + 6 {6, 6, 4} + 7 {7, 5, 4} + 8 {7, 6, 3} + 3 {7, 7, 2} \ + 3 {8, 4, 4} + 6 {8, 5, 3} + 6 {8, 6, 2} + 3 {8, 7, 1} + 6 {4, 4, 4, 4} + 21 {5, 4, 4, 3} + 22 {5, 5, 3, 3} + 21 {5, 5, 4, 2} + 5 {5, 5, 5, 1} + 21 {6, 4, 3, 3} + 30 {6, 4, 4, 2} + 45 {6, 5, 3, 2} + 24 {6, 5, 4, 1} + 21 {6, 6, 2, 2} + 18 {6, 6, 3, 1} + 5 {7, 3, 3, 3} + 24 {7, 4, 3, 2} + 14 {7, 4, 4, 1} + 18 {7, 5, 2, 2} + 27 {7, 5, 3, 1} + 19 {7, 6, 2, 1} + 6 {7, 7, 1, 1} + 3 {8, 3, 3, 2} + 6 {8, 4, 2, 2} + 7 {8, 4, 3, 1} + 8 {8, \ 5, 2, 1} + 3 {8, 6, 1, 1} + 5 {4, 3, 3, 3, 3} + 21 {4, 4, 3, 3, 2} + 22 {4, 4, 4, 2, \ 2} + 21 {4, 4, 4, 3, 1} + 22 {5, 3, 3, 3, 2} + 56 {5, 4, 3, 2, 2} + 48 {5, 4, 3, \ 3, 1} + 48 {5, 4, 4, 2, 1} + 20 {5, 5, 2, 2, 2} + 56 {5, 5, 3, 2, 1} + 22 {5, 5, 4, \ 1, 1} + 24 {6, 3, 3, 2, 2} + 18 {6, 3, 3, 3, 1} + 36 {6, 4, 2, 2, 2} + 72 {6, 4, 3, \ 2, 1} + 24 {6, 4, 4, 1, 1} + 48 {6, 5, 2, 2, 1} + 42 {6, 5, 3, 1, 1} + 18 {6, 6, 2, \ 1, 1} + 8 {7, 3, 2, 2, 2} + 16 {7, 3, 3, 2, 1} + 24 {7, 4, 2, 2, 1} + 24 {7, 4, 3, \ 1, 1} + 24 {7, 5, 2, 1, 1} + 8 {7, 6, 1, 1, 1} + {8, 2, 2, 2, 2} + 3 {8, 3, 2, 2, \ 1} + 2 {8, 3, 3, 1, 1} + 3 {8, 4, 2, 1, 1} + {8, 5, 1, 1, 1} + 6 {3, 3, 3, 3, 2, \ 2} + 3 {3, 3, 3, 3, 3, 1} + 18 {4, 3, 3, 2, 2, 2} + 24 {4, 3, 3, 3, 2, 1} + 21 {4, 4, 2, 2, 2, 2} + 45 {4, 4, 3, 2, 2, 1} + 30 {4, 4, 3, 3, 1, 1} + 21 {4, 4, 4, 2, 1, 1} + 18 {5, 3, 2, 2, 2, 2} + 42 {5, 3, 3, 2, 2, 1} + 24 {5, 3, 3, 3, 1, 1} + 48 {5, 4, 2, 2, 2, 1} + 72 {5, 4, 3, 2, 1, 1} + 18 {5, 4, 4, 1, 1, 1} + 36 {5, 5, 2, 2, 1, 1} + 24 {5, 5, 3, 1, 1, 1} + 6 {6, 2, 2, 2, 2, 2} + 24 {6, 3, 2, 2, 2, 1} + 30 {6, 3, 3, 2, 1, 1} + 36 {6, 4, 2, 2, 1, 1} + 30 {6, 4, 3, 1, 1, 1} + 24 {6, 5, 2, 1, 1, 1} + 6 {6, 6, 1, 1, 1, 1} + 3 {7, 2, 2, 2, 2, 1} + 9 {7, 3, 2, 2, 1, 1} + 6 {7, 3, 3, 1, 1, 1} + 9 {7, 4, 2, 1, 1, 1} + 3 {7, 5, 1, 1, 1, 1} + 3 {3, 3, 2, 2, 2, 2, 2} + 8 {3, 3, 3, 2, 2, 2, 1} + 7 {3, 3, 3, 3, 2, 1, 1} \ + 6 {4, 2, 2, 2, 2, 2, 2} + 19 {4, 3, 2, 2, 2, 2, 1} + 27 {4, 3, 3, 2, 2, 1, \ 1} + 14 {4, 3, 3, 3, 1, 1, 1} + 18 {4, 4, 2, 2, 2, 1, 1} + 24 {4, 4, 3, 2, 1, 1, \ 1} + 5 {4, 4, 4, 1, 1, 1, 1} + 8 {5, 2, 2, 2, 2, 2, 1} + 24 {5, 3, 2, 2, 2, 1, \ 1} + 24 {5, 3, 3, 2, 1, 1, 1} + 24 {5, 4, 2, 2, 1, 1, 1} + 16 {5, 4, 3, 1, 1, 1, \ 1} + 8 {5, 5, 2, 1, 1, 1, 1} + 3 {6, 2, 2, 2, 2, 1, 1} + 9 {6, 3, 2, 2, 1, 1, 1} \ + 6 {6, 3, 3, 1, 1, 1, 1} + 9 {6, 4, 2, 1, 1, 1, 1} + 3 {6, 5, 1, 1, 1, 1, 1} \ + {2, 2, 2, 2, 2, 2, 2, 2} + 3 {3, 2, 2, 2, 2, 2, 2, 1} + 6 {3, 3, 2, 2, 2, \ 2, 1, 1} + 6 {3, 3, 3, 2, 2, 1, 1, 1} + 3 {3, 3, 3, 3, 1, 1, 1, 1} + 3 {4, 2, 2, 2, 2, \ 2, 1, 1} + 8 {4, 3, 2, 2, 2, 1, 1, 1} + 7 {4, 3, 3, 2, 1, 1, 1, 1} + 6 {4, 4, 2, 2, 1, \ 1, 1, 1} + 3 {4, 4, 3, 1, 1, 1, 1, 1} + {5, 2, 2, 2, 2, 1, 1, 1} + 3 {5, 3, 2, 2, 1, \ 1, 1, 1} + 2 {5, 3, 3, 1, 1, 1, 1, 1} + 3 {5, 4, 2, 1, 1, 1, 1, 1} + {5, 5, 1, 1, 1, \ 1, 1, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pleth22m4 = Plethysm[riemsym,vier]", "Input"], Cell[OutputFormData["\<\ Parti[8, 8] + Parti[6, 6, 4] + Parti[8, 4, 4] + Parti[8, 6, 2] + 2*Parti[4, \ 4, 4, 4] + 3*Parti[5, 5, 3, 3] + 3*Parti[6, 4, 4, 2] + Parti[6, 5, 3, 2] + Parti[6, 5, \ 4, 1] + 3*Parti[6, 6, 2, 2] + Parti[7, 4, 3, 2] + 2*Parti[7, 5, 3, 1] + Parti[7, 6, \ 2, 1] + Parti[7, 7, 1, 1] + Parti[8, 4, 2, 2] + 3*Parti[4, 4, 4, 2, 2] + 2*Parti[5, 3, 3, 3, 2] + 2*Parti[5, 4, 3, 2, 2] + 2*Parti[5, 4, 3, 3, 1] + 2*Parti[5, 4, 4, 2, 1] + 2*Parti[5, 5, 3, 2, 1] + 2*Parti[5, 5, 4, 1, 1] + Parti[6, 3, 3, 3, 1] + 3*Parti[6, 4, 2, 2, 2] + 3*Parti[6, 4, 3, 2, 1] + 2*Parti[6, 5, 2, 2, 1] + 2*Parti[6, 5, 3, 1, 1] + Parti[7, 3, 3, 2, 1] + Parti[7, 4, 2, 2, 1] + Parti[7, 4, 3, 1, 1] + Parti[7, 5, 2, 1, 1] + Parti[8, 2, 2, 2, 2] + Parti[3, 3, 3, 3, 2, 2] + Parti[4, 3, 3, 3, 2, 1] + 3*Parti[4, 4, 2, 2, 2, 2] + Parti[4, 4, 3, 2, 2, 1] + 3*Parti[4, 4, 3, 3, \ 1, 1] + 2*Parti[5, 3, 3, 2, 2, 1] + 2*Parti[5, 4, 2, 2, 2, 1] + 3*Parti[5, 4, 3, 2, \ 1, 1] + Parti[5, 4, 4, 1, 1, 1] + 3*Parti[5, 5, 2, 2, 1, 1] + Parti[6, 2, 2, 2, 2, \ 2] + Parti[6, 3, 2, 2, 2, 1] + 2*Parti[6, 3, 3, 2, 1, 1] + 2*Parti[6, 4, 3, 1, \ 1, 1] + Parti[6, 5, 2, 1, 1, 1] + Parti[6, 6, 1, 1, 1, 1] + Parti[7, 3, 2, 2, 1, 1] \ + Parti[4, 2, 2, 2, 2, 2, 2] + Parti[4, 3, 2, 2, 2, 2, 1] + 2*Parti[4, 3, 3, 2, 2, 1, 1] + Parti[4, 4, 3, 2, 1, 1, 1] + Parti[5, 3, 2, 2, 2, 1, 1] + Parti[5, 3, 3, 2, 1, 1, 1] + Parti[5, 4, 2, 2, \ 1, 1, 1] + Parti[5, 4, 3, 1, 1, 1, 1] + Parti[6, 4, 2, 1, 1, 1, 1] + Parti[2, 2, 2, 2, 2, 2, 2, 2] + Parti[3, 3, 2, 2, 2, 2, 1, 1] + Parti[3, 3, 3, 3, 1, 1, 1, 1] + Parti[4, 4, 2, 2, 1, 1, 1, 1] + Parti[5, 5, 1, 1, 1, 1, 1, 1]\ \>", "\<\ {8, 8} + {6, 6, 4} + {8, 4, 4} + {8, 6, 2} + 2 {4, 4, 4, 4} + 3 {5, 5, 3, 3} \ + 3 {6, 4, 4, 2} + {6, 5, 3, 2} + {6, 5, 4, 1} + 3 {6, 6, 2, 2} + {7, 4, 3, \ 2} + 2 {7, 5, 3, 1} + {7, 6, 2, 1} + {7, 7, 1, 1} + {8, 4, 2, 2} + 3 {4, 4, 4, \ 2, 2} + 2 {5, 3, 3, 3, 2} + 2 {5, 4, 3, 2, 2} + 2 {5, 4, 3, 3, 1} + 2 {5, 4, 4, 2, \ 1} + 2 {5, 5, 3, 2, 1} + 2 {5, 5, 4, 1, 1} + {6, 3, 3, 3, 1} + 3 {6, 4, 2, 2, 2} \ + 3 {6, 4, 3, 2, 1} + 2 {6, 5, 2, 2, 1} + 2 {6, 5, 3, 1, 1} + {7, 3, 3, 2, 1} \ + {7, 4, 2, 2, 1} + {7, 4, 3, 1, 1} + {7, 5, 2, 1, 1} + {8, 2, 2, 2, 2} + {3, 3, 3, 3, 2, 2} + {4, 3, 3, 3, 2, 1} + 3 {4, 4, 2, 2, 2, 2} + {4, 4, 3, \ 2, 2, 1} + 3 {4, 4, 3, 3, 1, 1} + 2 {5, 3, 3, 2, 2, 1} + 2 {5, 4, 2, 2, 2, 1} + 3 {5, 4, 3, 2, 1, 1} + {5, 4, 4, 1, 1, 1} + 3 {5, 5, 2, 2, 1, 1} + {6, 2, 2, 2, 2, 2} + {6, 3, 2, 2, 2, 1} + 2 {6, 3, 3, 2, 1, 1} + 2 {6, 4, 3, 1, 1, 1} + {6, 5, 2, 1, 1, 1} + {6, 6, 1, 1, 1, 1} + {7, 3, 2, \ 2, 1, 1} + {4, 2, 2, 2, 2, 2, 2} + {4, 3, 2, 2, 2, 2, 1} + 2 {4, 3, 3, 2, 2, 1, 1} + {4, 4, 3, 2, 1, 1, 1} + {5, 3, 2, 2, 2, 1, 1} + {5, 3, 3, 2, 1, 1, 1} + {5, 4, 2, 2, 1, 1, 1} + {5, 4, 3, 1, 1, 1, 1} + {6, 4, 2, 1, 1, 1, 1} + {2, 2, 2, 2, 2, 2, 2, 2} + {3, 3, 2, 2, 2, 2, 1, 1} + {3, 3, 3, 3, 1, 1, 1, \ 1} + {4, 4, 2, 2, 1, 1, 1, 1} + {5, 5, 1, 1, 1, 1, 1, 1}\ \>"], "Output"] }, Open ]], Cell[TextData[{ "The number of minimal left ideals which occur in a decomposition of the \ full ring \[DoubleStruckCapitalC][", Cell[BoxData[ \(TraditionalForm\`S\_16\)]], "] is ", "46206736", Cell["", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["part16 = AllPartitions[16]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 1, 1, 1, 1, \ 1, 1], Parti[2, 2, 2, 2, 2, 2, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 2], Parti[3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1], Parti[3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 2, 2, 2, 1, 1, 1, 1, 1, \ 1, 1], Parti[3, 2, 2, 2, 2, 1, 1, 1, 1, 1], Parti[3, 2, 2, 2, 2, 2, 1, 1, 1], Parti[3, 2, 2, 2, 2, 2, 2, 1], Parti[3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 3, 2, 2, 1, 1, 1, 1, 1, \ 1], Parti[3, 3, 2, 2, 2, 1, 1, 1, 1], Parti[3, 3, 2, 2, 2, 2, 1, 1], Parti[3, 3, 2, 2, 2, 2, 2], Parti[3, 3, 3, 1, 1, 1, 1, 1, 1, 1], Parti[3, 3, 3, 2, 1, 1, 1, 1, 1], Parti[3, 3, 3, 2, 2, 1, 1, 1], Parti[3, 3, 3, 2, 2, 2, 1], Parti[3, 3, 3, 3, 1, 1, 1, 1], Parti[3, 3, 3, \ 3, 2, 1, 1], Parti[3, 3, 3, 3, 2, 2], Parti[3, 3, 3, 3, 3, 1], Parti[4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 2, 2, 1, 1, 1, 1, 1, 1, \ 1, 1], Parti[4, 2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[4, 2, 2, 2, 2, 1, 1, 1, 1], Parti[4, 2, 2, 2, 2, 2, 1, 1], Parti[4, 2, 2, 2, 2, 2, 2], Parti[4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 3, 2, 1, 1, 1, 1, 1, 1, \ 1], Parti[4, 3, 2, 2, 1, 1, 1, 1, 1], Parti[4, 3, 2, 2, 2, 1, 1, 1], Parti[4, 3, 2, 2, 2, 2, 1], Parti[4, 3, 3, 1, 1, 1, 1, 1, 1], Parti[4, 3, 3, 2, 1, 1, 1, 1], Parti[4, 3, 3, 2, 2, 1, 1], Parti[4, 3, 3, \ 2, 2, 2], Parti[4, 3, 3, 3, 1, 1, 1], Parti[4, 3, 3, 3, 2, 1], Parti[4, 3, 3, 3, 3], Parti[4, 4, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 4, 2, 1, 1, 1, 1, 1, 1], Parti[4, 4, 2, 2, 1, 1, 1, 1], Parti[4, 4, 2, 2, 2, 1, 1], Parti[4, 4, 2, \ 2, 2, 2], Parti[4, 4, 3, 1, 1, 1, 1, 1], Parti[4, 4, 3, 2, 1, 1, 1], Parti[4, 4, 3, \ 2, 2, 1], Parti[4, 4, 3, 3, 1, 1], Parti[4, 4, 3, 3, 2], Parti[4, 4, 4, 1, 1, 1, 1], Parti[4, 4, 4, 2, 1, 1], Parti[4, 4, 4, 2, 2], Parti[4, 4, 4, 3, 1], Parti[4, 4, 4, 4], Parti[5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 2, 2, 1, 1, 1, 1, 1, 1, \ 1], Parti[5, 2, 2, 2, 1, 1, 1, 1, 1], Parti[5, 2, 2, 2, 2, 1, 1, 1], Parti[5, 2, 2, 2, 2, 2, 1], Parti[5, 3, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 3, 2, 1, 1, 1, 1, 1, 1], Parti[5, 3, 2, 2, 1, 1, 1, 1], Parti[5, 3, 2, 2, 2, 1, 1], Parti[5, 3, 2, 2, 2, 2], Parti[5, 3, 3, 1, 1, \ 1, 1, 1], Parti[5, 3, 3, 2, 1, 1, 1], Parti[5, 3, 3, 2, 2, 1], Parti[5, 3, 3, 3, 1, \ 1], Parti[5, 3, 3, 3, 2], Parti[5, 4, 1, 1, 1, 1, 1, 1, 1], Parti[5, 4, 2, 1, \ 1, 1, 1, 1], Parti[5, 4, 2, 2, 1, 1, 1], Parti[5, 4, 2, 2, 2, 1], Parti[5, 4, 3, 1, 1, \ 1, 1], Parti[5, 4, 3, 2, 1, 1], Parti[5, 4, 3, 2, 2], Parti[5, 4, 3, 3, 1], Parti[5, 4, 4, 1, 1, 1], Parti[5, 4, 4, 2, 1], Parti[5, 4, 4, 3], Parti[5, 5, 1, 1, 1, 1, 1, 1], Parti[5, 5, 2, 1, 1, 1, 1], Parti[5, 5, 2, \ 2, 1, 1], Parti[5, 5, 2, 2, 2], Parti[5, 5, 3, 1, 1, 1], Parti[5, 5, 3, 2, 1], Parti[5, 5, 3, 3], Parti[5, 5, 4, 1, 1], Parti[5, 5, 4, 2], Parti[5, 5, 5, \ 1], Parti[6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[6, 2, 1, 1, 1, 1, 1, 1, 1, \ 1], Parti[6, 2, 2, 1, 1, 1, 1, 1, 1], Parti[6, 2, 2, 2, 1, 1, 1, 1], Parti[6, 2, 2, 2, 2, 1, 1], Parti[6, 2, 2, 2, 2, 2], Parti[6, 3, 1, 1, 1, \ 1, 1, 1, 1], Parti[6, 3, 2, 1, 1, 1, 1, 1], Parti[6, 3, 2, 2, 1, 1, 1], Parti[6, 3, 2, \ 2, 2, 1], Parti[6, 3, 3, 1, 1, 1, 1], Parti[6, 3, 3, 2, 1, 1], Parti[6, 3, 3, 2, 2], Parti[6, 3, 3, 3, 1], Parti[6, 4, 1, 1, 1, 1, 1, 1], Parti[6, 4, 2, 1, 1, \ 1, 1], Parti[6, 4, 2, 2, 1, 1], Parti[6, 4, 2, 2, 2], Parti[6, 4, 3, 1, 1, 1], Parti[6, 4, 3, 2, 1], Parti[6, 4, 3, 3], Parti[6, 4, 4, 1, 1], Parti[6, 4, \ 4, 2], Parti[6, 5, 1, 1, 1, 1, 1], Parti[6, 5, 2, 1, 1, 1], Parti[6, 5, 2, 2, 1], Parti[6, 5, 3, 1, 1], Parti[6, 5, 3, 2], Parti[6, 5, 4, 1], Parti[6, 5, 5], \ Parti[6, 6, 1, 1, 1, 1], Parti[6, 6, 2, 1, 1], Parti[6, 6, 2, 2], Parti[6, \ 6, 3, 1], Parti[6, 6, 4], Parti[7, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[7, 2, 1, 1, 1, \ 1, 1, 1, 1], Parti[7, 2, 2, 1, 1, 1, 1, 1], Parti[7, 2, 2, 2, 1, 1, 1], Parti[7, 2, 2, \ 2, 2, 1], Parti[7, 3, 1, 1, 1, 1, 1, 1], Parti[7, 3, 2, 1, 1, 1, 1], Parti[7, 3, 2, \ 2, 1, 1], Parti[7, 3, 2, 2, 2], Parti[7, 3, 3, 1, 1, 1], Parti[7, 3, 3, 2, 1], Parti[7, 3, 3, 3], Parti[7, 4, 1, 1, 1, 1, 1], Parti[7, 4, 2, 1, 1, 1], Parti[7, 4, 2, 2, 1], Parti[7, 4, 3, 1, 1], Parti[7, 4, 3, 2], Parti[7, 4, \ 4, 1], Parti[7, 5, 1, 1, 1, 1], Parti[7, 5, 2, 1, 1], Parti[7, 5, 2, 2], Parti[7, \ 5, 3, 1], Parti[7, 5, 4], Parti[7, 6, 1, 1, 1], Parti[7, 6, 2, 1], Parti[7, 6, 3], Parti[7, 7, 1, 1], Parti[7, 7, 2], Parti[8, 1, 1, 1, 1, 1, 1, 1, 1], Parti[8, 2, 1, 1, 1, 1, 1, 1], Parti[8, 2, 2, 1, 1, 1, 1], Parti[8, 2, 2, \ 2, 1, 1], Parti[8, 2, 2, 2, 2], Parti[8, 3, 1, 1, 1, 1, 1], Parti[8, 3, 2, 1, 1, 1], Parti[8, 3, 2, 2, 1], Parti[8, 3, 3, 1, 1], Parti[8, 3, 3, 2], Parti[8, 4, 1, 1, 1, 1], Parti[8, 4, 2, 1, 1], Parti[8, 4, 2, 2], Parti[8, \ 4, 3, 1], Parti[8, 4, 4], Parti[8, 5, 1, 1, 1], Parti[8, 5, 2, 1], Parti[8, 5, 3], Parti[8, 6, 1, 1], Parti[8, 6, 2], Parti[8, 7, 1], Parti[8, 8], Parti[9, 1, 1, 1, 1, 1, 1, 1], Parti[9, 2, 1, 1, 1, 1, 1], Parti[9, 2, 2, \ 1, 1, 1], Parti[9, 2, 2, 2, 1], Parti[9, 3, 1, 1, 1, 1], Parti[9, 3, 2, 1, 1], Parti[9, 3, 2, 2], Parti[9, 3, 3, 1], Parti[9, 4, 1, 1, 1], Parti[9, 4, 2, \ 1], Parti[9, 4, 3], Parti[9, 5, 1, 1], Parti[9, 5, 2], Parti[9, 6, 1], Parti[9, \ 7], Parti[10, 1, 1, 1, 1, 1, 1], Parti[10, 2, 1, 1, 1, 1], Parti[10, 2, 2, 1, \ 1], Parti[10, 2, 2, 2], Parti[10, 3, 1, 1, 1], Parti[10, 3, 2, 1], Parti[10, 3, \ 3], Parti[10, 4, 1, 1], Parti[10, 4, 2], Parti[10, 5, 1], Parti[10, 6], Parti[11, 1, 1, 1, 1, 1], Parti[11, 2, 1, 1, 1], Parti[11, 2, 2, 1], Parti[11, 3, 1, 1], Parti[11, 3, 2], Parti[11, 4, 1], Parti[11, 5], Parti[12, 1, 1, 1, 1], Parti[12, 2, 1, 1], Parti[12, 2, 2], Parti[12, 3, \ 1], Parti[12, 4], Parti[13, 1, 1, 1], Parti[13, 2, 1], Parti[13, 3], Parti[14, \ 1, 1], Parti[14, 2], Parti[15, 1], Parti[16]]\ \>", "\<\ {{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1}, {2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 1, 1}, {2, 2, 2, 2, \ 2, 2, 2, 2}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 2, 2, 1, 1, 1, 1, 1}, {3, 2, 2, 2, 2, 2, 1, 1, 1}, {3, 2, 2, 2, \ 2, 2, 2, 1}, {3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 2, 2, 1, 1, 1, 1, 1, 1}, {3, 3, 2, 2, 2, 1, 1, 1, 1}, {3, 3, 2, 2, \ 2, 2, 1, 1}, {3, 3, 2, 2, 2, 2, 2}, {3, 3, 3, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 3, 2, 1, 1, \ 1, 1, 1}, {3, 3, 3, 2, 2, 1, 1, 1}, {3, 3, 3, 2, 2, 2, 1}, {3, 3, 3, 3, 1, 1, 1, 1}, {3, 3, 3, 3, 2, 1, 1}, {3, 3, 3, 3, 2, 2}, {3, 3, 3, 3, 3, 1}, {4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1}, {4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 2, 2, 1, 1, 1, 1, 1, 1}, {4, 2, 2, 2, 2, 1, 1, 1, 1}, {4, 2, 2, 2, 2, 2, 1, 1}, {4, 2, 2, 2, 2, 2, \ 2}, {4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 3, 2, 1, 1, 1, 1, 1, 1, 1}, {4, 3, 2, 2, 1, 1, 1, 1, 1}, {4, 3, 2, 2, 2, 1, 1, 1}, {4, 3, 2, 2, 2, 2, \ 1}, {4, 3, 3, 1, 1, 1, 1, 1, 1}, {4, 3, 3, 2, 1, 1, 1, 1}, {4, 3, 3, 2, 2, 1, \ 1}, {4, 3, 3, 2, 2, 2}, {4, 3, 3, 3, 1, 1, 1}, {4, 3, 3, 3, 2, 1}, {4, 3, 3, 3, \ 3}, {4, 4, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 4, 2, 1, 1, 1, 1, 1, 1}, {4, 4, 2, 2, \ 1, 1, 1, 1}, {4, 4, 2, 2, 2, 1, 1}, {4, 4, 2, 2, 2, 2}, {4, 4, 3, 1, 1, 1, 1, 1}, {4, 4, 3, 2, 1, 1, 1}, {4, 4, 3, 2, 2, 1}, {4, 4, 3, 3, 1, 1}, {4, 4, 3, 3, \ 2}, {4, 4, 4, 1, 1, 1, 1}, {4, 4, 4, 2, 1, 1}, {4, 4, 4, 2, 2}, {4, 4, 4, 3, \ 1}, {4, 4, 4, 4}, {5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1, \ 1, 1, 1, 1}, {5, 2, 2, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 2, 2, 1, 1, 1, 1, 1}, {5, 2, 2, 2, \ 2, 1, 1, 1}, {5, 2, 2, 2, 2, 2, 1}, {5, 3, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 3, 2, 1, 1, 1, \ 1, 1, 1}, {5, 3, 2, 2, 1, 1, 1, 1}, {5, 3, 2, 2, 2, 1, 1}, {5, 3, 2, 2, 2, 2}, {5, 3, 3, 1, 1, 1, 1, 1}, {5, 3, 3, 2, 1, 1, 1}, {5, 3, 3, 2, 2, 1}, {5, 3, 3, 3, 1, 1}, {5, 3, 3, 3, 2}, {5, 4, 1, 1, 1, 1, 1, 1, 1}, {5, 4, 2, 1, 1, 1, 1, 1}, {5, 4, 2, 2, 1, 1, 1}, {5, 4, 2, 2, 2, 1}, {5, 4, 3, 1, 1, 1, 1}, {5, 4, 3, 2, 1, 1}, {5, 4, 3, 2, 2}, {5, 4, 3, 3, \ 1}, {5, 4, 4, 1, 1, 1}, {5, 4, 4, 2, 1}, {5, 4, 4, 3}, {5, 5, 1, 1, 1, 1, 1, \ 1}, {5, 5, 2, 1, 1, 1, 1}, {5, 5, 2, 2, 1, 1}, {5, 5, 2, 2, 2}, {5, 5, 3, 1, 1, \ 1}, {5, 5, 3, 2, 1}, {5, 5, 3, 3}, {5, 5, 4, 1, 1}, {5, 5, 4, 2}, {5, 5, 5, 1}, \ {6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {6, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {6, 2, 2, 1, 1, 1, 1, 1, 1}, {6, 2, 2, 2, 1, 1, 1, 1}, {6, 2, 2, 2, 2, 1, \ 1}, {6, 2, 2, 2, 2, 2}, {6, 3, 1, 1, 1, 1, 1, 1, 1}, {6, 3, 2, 1, 1, 1, 1, 1}, {6, 3, 2, 2, 1, 1, 1}, {6, 3, 2, 2, 2, 1}, {6, 3, 3, 1, 1, 1, 1}, {6, 3, 3, \ 2, 1, 1}, {6, 3, 3, 2, 2}, {6, 3, 3, 3, 1}, {6, 4, 1, 1, 1, 1, 1, 1}, {6, 4, 2, 1, 1, \ 1, 1}, {6, 4, 2, 2, 1, 1}, {6, 4, 2, 2, 2}, {6, 4, 3, 1, 1, 1}, {6, 4, 3, 2, 1}, {6, 4, 3, 3}, {6, 4, 4, 1, 1}, {6, 4, 4, 2}, {6, 5, 1, 1, 1, 1, 1}, {6, 5, 2, 1, 1, 1}, {6, 5, 2, 2, 1}, {6, 5, 3, 1, 1}, {6, 5, 3, 2}, {6, 5, \ 4, 1}, {6, 5, 5}, {6, 6, 1, 1, 1, 1}, {6, 6, 2, 1, 1}, {6, 6, 2, 2}, {6, 6, 3, 1}, \ {6, 6, 4}, {7, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {7, 2, 1, 1, 1, 1, 1, 1, 1}, {7, 2, 2, 1, \ 1, 1, 1, 1}, {7, 2, 2, 2, 1, 1, 1}, {7, 2, 2, 2, 2, 1}, {7, 3, 1, 1, 1, 1, 1, 1}, {7, 3, 2, 1, 1, 1, 1}, {7, 3, 2, 2, 1, 1}, {7, 3, 2, 2, 2}, {7, 3, 3, 1, 1, \ 1}, {7, 3, 3, 2, 1}, {7, 3, 3, 3}, {7, 4, 1, 1, 1, 1, 1}, {7, 4, 2, 1, 1, 1}, {7, 4, 2, 2, 1}, {7, 4, 3, 1, 1}, {7, 4, 3, 2}, {7, 4, 4, 1}, {7, 5, 1, 1, \ 1, 1}, {7, 5, 2, 1, 1}, {7, 5, 2, 2}, {7, 5, 3, 1}, {7, 5, 4}, {7, 6, 1, 1, 1}, \ {7, 6, 2, 1}, {7, 6, 3}, {7, 7, 1, 1}, {7, 7, 2}, {8, 1, 1, 1, 1, 1, 1, 1, 1}, {8, 2, 1, 1, 1, 1, 1, 1}, {8, 2, 2, 1, 1, 1, 1}, {8, 2, 2, 2, 1, 1}, {8, 2, \ 2, 2, 2}, {8, 3, 1, 1, 1, 1, 1}, {8, 3, 2, 1, 1, 1}, {8, 3, 2, 2, 1}, {8, 3, 3, 1, \ 1}, {8, 3, 3, 2}, {8, 4, 1, 1, 1, 1}, {8, 4, 2, 1, 1}, {8, 4, 2, 2}, {8, 4, 3, \ 1}, {8, 4, 4}, {8, 5, 1, 1, 1}, {8, 5, 2, 1}, {8, 5, 3}, {8, 6, 1, 1}, {8, 6, \ 2}, {8, 7, 1}, {8, 8}, {9, 1, 1, 1, 1, 1, 1, 1}, {9, 2, 1, 1, 1, 1, 1}, {9, 2, 2, 1, 1, 1}, {9, 2, 2, 2, 1}, {9, 3, 1, 1, 1, 1}, {9, 3, 2, 1, 1}, {9, 3, 2, 2}, {9, 3, 3, 1}, {9, 4, 1, 1, 1}, {9, 4, 2, 1}, {9, 4, 3}, {9, \ 5, 1, 1}, {9, 5, 2}, {9, 6, 1}, {9, 7}, {10, 1, 1, 1, 1, 1, 1}, {10, 2, 1, 1, 1, 1}, {10, 2, 2, 1, 1}, {10, 2, 2, 2}, {10, 3, 1, 1, 1}, {10, 3, 2, 1}, {10, 3, \ 3}, {10, 4, 1, 1}, {10, 4, 2}, {10, 5, 1}, {10, 6}, {11, 1, 1, 1, 1, 1}, {11, \ 2, 1, 1, 1}, {11, 2, 2, 1}, {11, 3, 1, 1}, {11, 3, 2}, {11, 4, 1}, {11, 5}, {12, 1, 1, \ 1, 1}, {12, 2, 1, 1}, {12, 2, 2}, {12, 3, 1}, {12, 4}, {13, 1, 1, 1}, {13, 2, 1}, \ {13, 3}, {14, 1, 1}, {14, 2}, {15, 1}, {16}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ part16", "Input"], Cell[OutputFormData["\<\ HoldList[1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430, 105, 896, 3900, \ 10752, 20020, 24960, 18018, 2640, 13860, 38220, 65520, 68640, 30030, 21840, 69888, \ 112112, 91520, 60060, 100100, 51480, 36036, 455, 4004, 16848, 43120, 71500, 73710, 32032, \ 14300, 71680, 180180, 266240, 210210, 112112, 318500, 416988, 200200, 240240, \ 292864, 60060, 23100, 114660, 262080, 320320, 150150, 206388, 512512, 500500, 336336, \ 231660, 140140, 300300, 171600, 180180, 24024, 1365, 11648, 45760, 105600, 150150, 116480, \ 42120, 194040, 429000, 515970, 240240, 280280, 673920, 648648, 411840, 280280, \ 80640, 360360, 698880, 630630, 582400, 1153152, 640640, 549120, 360360, 549120, 180180, \ 76440, 305760, 480480, 250250, 448448, 640640, 171600, 280280, 231660, 60060, \ 3003, 24024, 85800, 173250, 200200, 91728, 82368, 337920, 630630, 559104, 429000, \ 819000, 448448, 360360, 155232, 600600, 928746, 480480, 819000, 1153152, 300300, 411840, \ 336336, 168168, 559104, 630630, 648648, 500500, 292864, 36036, 91728, 240240, \ 150150, 200200, 51480, 5005, 36608, 115830, 197120, 168168, 114400, 404250, 600600, 305760, \ 429000, 582400, 140140, 197120, 630630, 698880, 673920, 512512, 240240, 200200, \ 515970, 320320, 416988, 100100, 116480, 210210, 91520, 32032, 30030, 6435, 42042, \ 114400, 155232, 76440, 115830, 337920, 360360, 280280, 206388, 173250, 429000, \ 262080, 318500, 60060, 150150, 266240, 112112, 73710, 68640, 18018, 1430, 6435, 36608, \ 82368, 80640, 85800, 194040, 114660, 112112, 105600, 180180, 69888, 71500, 65520, 24960, \ 3432, 5005, 24024, 42120, 23100, 45760, 71680, 21840, 43120, 38220, 20020, 3640, 3003, \ 11648, 14300, 16848, 13860, 10752, 2548, 1365, 4004, 2640, 3900, 1260, 455, 896, \ 440, 105, 104, 15, 1]\ \>", "\<\ {1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430, 105, 896, 3900, 10752, 20020, \ 24960, 18018, 2640, 13860, 38220, 65520, 68640, 30030, 21840, 69888, 112112, \ 91520, 60060, 100100, 51480, 36036, 455, 4004, 16848, 43120, 71500, 73710, 32032, 14300, \ 71680, 180180, 266240, 210210, 112112, 318500, 416988, 200200, 240240, 292864, \ 60060, 23100, 114660, 262080, 320320, 150150, 206388, 512512, 500500, 336336, 231660, \ 140140, 300300, 171600, 180180, 24024, 1365, 11648, 45760, 105600, 150150, 116480, \ 42120, 194040, 429000, 515970, 240240, 280280, 673920, 648648, 411840, 280280, \ 80640, 360360, 698880, 630630, 582400, 1153152, 640640, 549120, 360360, 549120, 180180, \ 76440, 305760, 480480, 250250, 448448, 640640, 171600, 280280, 231660, 60060, \ 3003, 24024, 85800, 173250, 200200, 91728, 82368, 337920, 630630, 559104, 429000, \ 819000, 448448, 360360, 155232, 600600, 928746, 480480, 819000, 1153152, 300300, 411840, \ 336336, 168168, 559104, 630630, 648648, 500500, 292864, 36036, 91728, 240240, \ 150150, 200200, 51480, 5005, 36608, 115830, 197120, 168168, 114400, 404250, 600600, 305760, \ 429000, 582400, 140140, 197120, 630630, 698880, 673920, 512512, 240240, 200200, \ 515970, 320320, 416988, 100100, 116480, 210210, 91520, 32032, 30030, 6435, 42042, \ 114400, 155232, 76440, 115830, 337920, 360360, 280280, 206388, 173250, 429000, \ 262080, 318500, 60060, 150150, 266240, 112112, 73710, 68640, 18018, 1430, 6435, 36608, \ 82368, 80640, 85800, 194040, 114660, 112112, 105600, 180180, 69888, 71500, 65520, 24960, \ 3432, 5005, 24024, 42120, 23100, 45760, 71680, 21840, 43120, 38220, 20020, 3640, 3003, \ 11648, 14300, 16848, 13860, 10752, 2548, 1365, 4004, 2640, 3900, 1260, 455, 896, \ 440, 105, 104, 15, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["allideals = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 46206736\ \>", "\<\ 46206736\ \>"], "Output"] }, Open ]], Cell["\<\ We determine the number of such ideals lying in equivalence classes of \ minimal left ideals which do not occur in lr22m22m22m22 or pleth22m4.\ \>", "Text"], Cell["Ad lr22m22m22m22:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ lr22m22m22m22", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8, 8], 3*Parti[6, 5, 5], 6*Parti[6, 6, 4], 7*Parti[7, 5, 4], 8*Parti[7, 6, 3], 3*Parti[7, 7, 2], 3*Parti[8, 4, 4], 6*Parti[8, 5, 3], 6*Parti[8, 6, 2], 3*Parti[8, 7, 1], 6*Parti[4, 4, 4, 4], 21*Parti[5, 4, 4, \ 3], 22*Parti[5, 5, 3, 3], 21*Parti[5, 5, 4, 2], 5*Parti[5, 5, 5, 1], \ 21*Parti[6, 4, 3, 3], 30*Parti[6, 4, 4, 2], 45*Parti[6, 5, 3, 2], 24*Parti[6, 5, 4, 1], 21*Parti[6, 6, 2, 2], 18*Parti[6, 6, 3, 1], 5*Parti[7, 3, 3, 3], \ 24*Parti[7, 4, 3, 2], 14*Parti[7, 4, 4, 1], 18*Parti[7, 5, 2, 2], 27*Parti[7, 5, 3, 1], 19*Parti[7, 6, 2, 1], 6*Parti[7, 7, 1, 1], 3*Parti[8, 3, 3, 2], 6*Parti[8, \ 4, 2, 2], 7*Parti[8, 4, 3, 1], 8*Parti[8, 5, 2, 1], 3*Parti[8, 6, 1, 1], 5*Parti[4, \ 3, 3, 3, 3], 21*Parti[4, 4, 3, 3, 2], 22*Parti[4, 4, 4, 2, 2], 21*Parti[4, 4, 4, 3, 1], 22*Parti[5, 3, 3, 3, 2], 56*Parti[5, 4, 3, 2, 2], 48*Parti[5, 4, 3, 3, 1], 48*Parti[5, 4, 4, 2, 1], 20*Parti[5, 5, 2, 2, 2], 56*Parti[5, 5, 3, 2, 1], 22*Parti[5, 5, 4, 1, 1], 24*Parti[6, 3, 3, 2, 2], 18*Parti[6, 3, 3, 3, 1], 36*Parti[6, 4, 2, 2, 2], 72*Parti[6, 4, 3, 2, 1], 24*Parti[6, 4, 4, 1, 1], 48*Parti[6, 5, 2, 2, 1], 42*Parti[6, 5, 3, 1, 1], 18*Parti[6, 6, 2, 1, 1], 8*Parti[7, 3, 2, 2, 2], 16*Parti[7, 3, 3, 2, 1], 24*Parti[7, 4, 2, 2, 1], 24*Parti[7, 4, 3, 1, 1], 24*Parti[7, 5, 2, 1, 1], 8*Parti[7, 6, 1, 1, 1], Parti[8, 2, 2, 2, 2], 3*Parti[8, 3, 2, 2, 1], 2*Parti[8, 3, 3, 1, 1], 3*Parti[8, 4, 2, 1, 1], Parti[8, 5, 1, 1, 1], 6*Parti[3, 3, 3, 3, 2, 2], 3*Parti[3, 3, 3, 3, 3, 1], 18*Parti[4, 3, 3, 2, 2, 2], 24*Parti[4, 3, 3, 3, \ 2, 1], 21*Parti[4, 4, 2, 2, 2, 2], 45*Parti[4, 4, 3, 2, 2, 1], 30*Parti[4, 4, 3, \ 3, 1, 1], 21*Parti[4, 4, 4, 2, 1, 1], 18*Parti[5, 3, 2, 2, 2, 2], 42*Parti[5, 3, 3, \ 2, 2, 1], 24*Parti[5, 3, 3, 3, 1, 1], 48*Parti[5, 4, 2, 2, 2, 1], 72*Parti[5, 4, 3, \ 2, 1, 1], 18*Parti[5, 4, 4, 1, 1, 1], 36*Parti[5, 5, 2, 2, 1, 1], 24*Parti[5, 5, 3, \ 1, 1, 1], 6*Parti[6, 2, 2, 2, 2, 2], 24*Parti[6, 3, 2, 2, 2, 1], 30*Parti[6, 3, 3, 2, \ 1, 1], 36*Parti[6, 4, 2, 2, 1, 1], 30*Parti[6, 4, 3, 1, 1, 1], 24*Parti[6, 5, 2, \ 1, 1, 1], 6*Parti[6, 6, 1, 1, 1, 1], 3*Parti[7, 2, 2, 2, 2, 1], 9*Parti[7, 3, 2, 2, \ 1, 1], 6*Parti[7, 3, 3, 1, 1, 1], 9*Parti[7, 4, 2, 1, 1, 1], 3*Parti[7, 5, 1, 1, \ 1, 1], 3*Parti[3, 3, 2, 2, 2, 2, 2], 8*Parti[3, 3, 3, 2, 2, 2, 1], 7*Parti[3, 3, 3, 3, 2, 1, 1], 6*Parti[4, 2, 2, 2, 2, 2, 2], 19*Parti[4, 3, 2, 2, 2, 2, 1], 27*Parti[4, 3, 3, 2, 2, 1, 1], 14*Parti[4, 3, 3, 3, 1, 1, 1], 18*Parti[4, 4, 2, 2, 2, 1, 1], 24*Parti[4, 4, 3, 2, 1, 1, 1], 5*Parti[4, 4, 4, 1, 1, 1, 1], 8*Parti[5, 2, 2, 2, 2, 2, 1], 24*Parti[5, 3, 2, 2, 2, 1, 1], 24*Parti[5, 3, 3, 2, 1, 1, 1], 24*Parti[5, 4, 2, 2, 1, 1, 1], 16*Parti[5, 4, 3, 1, 1, 1, 1], 8*Parti[5, 5, 2, 1, 1, 1, 1], 3*Parti[6, 2, 2, 2, 2, 1, 1], 9*Parti[6, 3, 2, 2, 1, 1, 1], 6*Parti[6, 3, 3, 1, 1, 1, 1], 9*Parti[6, 4, 2, 1, 1, 1, 1], 3*Parti[6, 5, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 2], 3*Parti[3, 2, 2, 2, 2, 2, 2, 1], 6*Parti[3, 3, 2, 2, 2, 2, 1, 1], 6*Parti[3, 3, 3, 2, 2, 1, 1, 1], 3*Parti[3, 3, 3, 3, 1, 1, 1, 1], 3*Parti[4, 2, 2, 2, 2, 2, 1, 1], 8*Parti[4, 3, 2, 2, 2, 1, 1, 1], 7*Parti[4, 3, 3, 2, 1, 1, 1, 1], 6*Parti[4, 4, 2, 2, 1, 1, 1, 1], 3*Parti[4, 4, 3, 1, 1, 1, 1, 1], Parti[5, 2, 2, 2, 2, 1, 1, 1], 3*Parti[5, 3, 2, 2, 1, 1, 1, 1], 2*Parti[5, 3, 3, 1, 1, 1, 1, 1], 3*Parti[5, 4, 2, 1, 1, 1, 1, 1], Parti[5, 5, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8, 8}, 3 {6, 5, 5}, 6 {6, 6, 4}, 7 {7, 5, 4}, 8 {7, 6, 3}, 3 {7, 7, 2}, 3 \ {8, 4, 4}, 6 {8, 5, 3}, 6 {8, 6, 2}, 3 {8, 7, 1}, 6 {4, 4, 4, 4}, 21 {5, 4, 4, 3}, 22 {5, 5, 3, 3}, 21 {5, 5, 4, 2}, 5 {5, 5, 5, 1}, 21 {6, 4, 3, 3}, 30 {6, \ 4, 4, 2}, 45 {6, 5, 3, 2}, 24 {6, 5, 4, 1}, 21 {6, 6, 2, 2}, 18 {6, 6, 3, 1}, 5 {7, \ 3, 3, 3}, 24 {7, 4, 3, 2}, 14 {7, 4, 4, 1}, 18 {7, 5, 2, 2}, 27 {7, 5, 3, 1}, 19 {7, \ 6, 2, 1}, 6 {7, 7, 1, 1}, 3 {8, 3, 3, 2}, 6 {8, 4, 2, 2}, 7 {8, 4, 3, 1}, 8 {8, 5, 2, \ 1}, 3 {8, 6, 1, 1}, 5 {4, 3, 3, 3, 3}, 21 {4, 4, 3, 3, 2}, 22 {4, 4, 4, 2, 2}, 21 {4, 4, 4, 3, 1}, 22 {5, 3, 3, 3, 2}, 56 {5, 4, 3, 2, 2}, 48 {5, 4, 3, 3, \ 1}, 48 {5, 4, 4, 2, 1}, 20 {5, 5, 2, 2, 2}, 56 {5, 5, 3, 2, 1}, 22 {5, 5, 4, 1, \ 1}, 24 {6, 3, 3, 2, 2}, 18 {6, 3, 3, 3, 1}, 36 {6, 4, 2, 2, 2}, 72 {6, 4, 3, 2, \ 1}, 24 {6, 4, 4, 1, 1}, 48 {6, 5, 2, 2, 1}, 42 {6, 5, 3, 1, 1}, 18 {6, 6, 2, 1, \ 1}, 8 {7, 3, 2, 2, 2}, 16 {7, 3, 3, 2, 1}, 24 {7, 4, 2, 2, 1}, 24 {7, 4, 3, 1, \ 1}, 24 {7, 5, 2, 1, 1}, 8 {7, 6, 1, 1, 1}, {8, 2, 2, 2, 2}, 3 {8, 3, 2, 2, 1}, 2 {8, 3, 3, 1, 1}, 3 {8, 4, 2, 1, 1}, {8, 5, 1, 1, 1}, 6 {3, 3, 3, 3, 2, \ 2}, 3 {3, 3, 3, 3, 3, 1}, 18 {4, 3, 3, 2, 2, 2}, 24 {4, 3, 3, 3, 2, 1}, 21 {4, 4, 2, 2, 2, 2}, 45 {4, 4, 3, 2, 2, 1}, 30 {4, 4, 3, 3, 1, 1}, 21 {4, 4, 4, 2, 1, 1}, 18 {5, 3, 2, 2, 2, 2}, 42 {5, 3, 3, 2, 2, 1}, 24 {5, 3, 3, 3, 1, 1}, 48 {5, 4, 2, 2, 2, 1}, 72 {5, 4, 3, 2, 1, 1}, 18 {5, 4, 4, 1, 1, 1}, 36 {5, 5, 2, 2, 1, 1}, 24 {5, 5, 3, 1, 1, 1}, 6 {6, 2, 2, 2, 2, 2}, 24 {6, 3, 2, 2, 2, 1}, 30 {6, 3, 3, 2, 1, 1}, 36 {6, 4, 2, 2, 1, 1}, 30 {6, 4, 3, 1, 1, 1}, 24 {6, 5, 2, 1, 1, 1}, 6 {6, 6, 1, 1, 1, 1}, 3 {7, 2, 2, 2, 2, 1}, 9 {7, 3, 2, 2, 1, 1}, 6 {7, 3, 3, 1, 1, 1}, 9 {7, 4, 2, 1, 1, 1}, 3 {7, 5, 1, 1, 1, 1}, 3 {3, 3, 2, 2, 2, 2, 2}, 8 {3, 3, 3, 2, 2, 2, 1}, 7 {3, 3, 3, 3, 2, 1, 1}, 6 {4, 2, 2, 2, 2, 2, 2}, 19 {4, 3, 2, 2, 2, 2, 1}, 27 {4, 3, 3, 2, 2, 1, \ 1}, 14 {4, 3, 3, 3, 1, 1, 1}, 18 {4, 4, 2, 2, 2, 1, 1}, 24 {4, 4, 3, 2, 1, 1, \ 1}, 5 {4, 4, 4, 1, 1, 1, 1}, 8 {5, 2, 2, 2, 2, 2, 1}, 24 {5, 3, 2, 2, 2, 1, 1}, \ 24 {5, 3, 3, 2, 1, 1, 1}, 24 {5, 4, 2, 2, 1, 1, 1}, 16 {5, 4, 3, 1, 1, 1, \ 1}, 8 {5, 5, 2, 1, 1, 1, 1}, 3 {6, 2, 2, 2, 2, 1, 1}, 9 {6, 3, 2, 2, 1, 1, 1}, 6 {6, 3, 3, 1, 1, 1, 1}, 9 {6, 4, 2, 1, 1, 1, 1}, 3 {6, 5, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 2}, 3 {3, 2, 2, 2, 2, 2, 2, 1}, 6 {3, 3, 2, 2, 2, 2, \ 1, 1}, 6 {3, 3, 3, 2, 2, 1, 1, 1}, 3 {3, 3, 3, 3, 1, 1, 1, 1}, 3 {4, 2, 2, 2, 2, \ 2, 1, 1}, 8 {4, 3, 2, 2, 2, 1, 1, 1}, 7 {4, 3, 3, 2, 1, 1, 1, 1}, 6 {4, 4, 2, 2, 1, \ 1, 1, 1}, 3 {4, 4, 3, 1, 1, 1, 1, 1}, {5, 2, 2, 2, 2, 1, 1, 1}, 3 {5, 3, 2, 2, 1, 1, \ 1, 1}, 2 {5, 3, 3, 1, 1, 1, 1, 1}, 3 {5, 4, 2, 1, 1, 1, 1, 1}, {5, 5, 1, 1, 1, 1, \ 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 127\ \>", "\<\ 127\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["list = list /. _*x_Parti :> x", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8, 8], Parti[6, 5, 5], Parti[6, 6, 4], Parti[7, 5, 4], \ Parti[7, 6, 3], Parti[7, 7, 2], Parti[8, 4, 4], Parti[8, 5, 3], Parti[8, 6, 2], Parti[8, 7, \ 1], Parti[4, 4, 4, 4], Parti[5, 4, 4, 3], Parti[5, 5, 3, 3], Parti[5, 5, 4, 2], \ Parti[5, 5, 5, 1], Parti[6, 4, 3, 3], Parti[6, 4, 4, 2], Parti[6, 5, 3, 2], \ Parti[6, 5, 4, 1], Parti[6, 6, 2, 2], Parti[6, 6, 3, 1], Parti[7, 3, 3, 3], \ Parti[7, 4, 3, 2], Parti[7, 4, 4, 1], Parti[7, 5, 2, 2], Parti[7, 5, 3, 1], \ Parti[7, 6, 2, 1], Parti[7, 7, 1, 1], Parti[8, 3, 3, 2], Parti[8, 4, 2, 2], \ Parti[8, 4, 3, 1], Parti[8, 5, 2, 1], Parti[8, 6, 1, 1], Parti[4, 3, 3, 3, \ 3], Parti[4, 4, 3, 3, 2], Parti[4, 4, 4, 2, 2], Parti[4, 4, 4, 3, 1], Parti[5, 3, 3, 3, 2], Parti[5, 4, 3, 2, 2], Parti[5, 4, 3, 3, 1], Parti[5, 4, 4, 2, 1], Parti[5, 5, 2, 2, 2], Parti[5, 5, 3, 2, 1], Parti[5, 5, 4, 1, 1], Parti[6, 3, 3, 2, 2], Parti[6, 3, 3, 3, 1], Parti[6, 4, 2, 2, 2], Parti[6, 4, 3, 2, 1], Parti[6, 4, 4, 1, 1], Parti[6, 5, 2, 2, 1], Parti[6, 5, 3, 1, 1], Parti[6, 6, 2, 1, 1], Parti[7, 3, 2, 2, 2], Parti[7, 3, 3, 2, 1], Parti[7, 4, 2, 2, 1], Parti[7, 4, 3, 1, 1], Parti[7, 5, 2, 1, 1], Parti[7, 6, 1, 1, 1], Parti[8, 2, 2, 2, 2], Parti[8, 3, 2, 2, 1], Parti[8, 3, 3, 1, 1], Parti[8, 4, 2, 1, 1], Parti[8, 5, 1, 1, 1], Parti[3, 3, 3, 3, 2, 2], Parti[3, 3, 3, 3, 3, 1], Parti[4, 3, 3, 2, 2, 2], Parti[4, 3, 3, 3, 2, 1], Parti[4, 4, 2, 2, 2, 2], Parti[4, 4, 3, 2, 2, 1], Parti[4, 4, 3, 3, 1, 1], Parti[4, 4, 4, 2, 1, 1], Parti[5, 3, 2, 2, 2, 2], Parti[5, 3, 3, 2, 2, 1], Parti[5, 3, 3, 3, 1, 1], Parti[5, 4, 2, 2, 2, 1], Parti[5, 4, 3, 2, 1, 1], Parti[5, 4, 4, 1, 1, 1], Parti[5, 5, 2, 2, 1, 1], Parti[5, 5, 3, 1, 1, 1], Parti[6, 2, 2, 2, 2, 2], Parti[6, 3, 2, 2, 2, 1], Parti[6, 3, 3, 2, 1, 1], Parti[6, 4, 2, 2, 1, 1], Parti[6, 4, 3, 1, 1, 1], Parti[6, 5, 2, 1, 1, 1], Parti[6, 6, 1, 1, 1, 1], Parti[7, 2, 2, 2, 2, 1], Parti[7, 3, 2, 2, 1, 1], Parti[7, 3, 3, 1, 1, 1], Parti[7, 4, 2, 1, 1, 1], Parti[7, 5, 1, 1, 1, 1], Parti[3, 3, 2, 2, 2, 2, 2], Parti[3, 3, 3, 2, 2, 2, 1], Parti[3, 3, 3, 3, \ 2, 1, 1], Parti[4, 2, 2, 2, 2, 2, 2], Parti[4, 3, 2, 2, 2, 2, 1], Parti[4, 3, 3, 2, \ 2, 1, 1], Parti[4, 3, 3, 3, 1, 1, 1], Parti[4, 4, 2, 2, 2, 1, 1], Parti[4, 4, 3, 2, \ 1, 1, 1], Parti[4, 4, 4, 1, 1, 1, 1], Parti[5, 2, 2, 2, 2, 2, 1], Parti[5, 3, 2, 2, \ 2, 1, 1], Parti[5, 3, 3, 2, 1, 1, 1], Parti[5, 4, 2, 2, 1, 1, 1], Parti[5, 4, 3, 1, \ 1, 1, 1], Parti[5, 5, 2, 1, 1, 1, 1], Parti[6, 2, 2, 2, 2, 1, 1], Parti[6, 3, 2, 2, \ 1, 1, 1], Parti[6, 3, 3, 1, 1, 1, 1], Parti[6, 4, 2, 1, 1, 1, 1], Parti[6, 5, 1, 1, \ 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 2], Parti[3, 2, 2, 2, 2, 2, 2, 1], Parti[3, 3, 2, 2, 2, 2, 1, 1], Parti[3, 3, 3, 2, 2, 1, 1, 1], Parti[3, 3, 3, 3, 1, 1, 1, 1], Parti[4, 2, 2, 2, 2, 2, 1, 1], Parti[4, 3, 2, 2, 2, 1, 1, 1], Parti[4, 3, 3, 2, 1, 1, 1, 1], Parti[4, 4, 2, 2, 1, 1, 1, 1], Parti[4, 4, 3, 1, 1, 1, 1, 1], Parti[5, 2, 2, 2, 2, 1, 1, 1], Parti[5, 3, 2, 2, 1, 1, 1, 1], Parti[5, 3, 3, 1, 1, 1, 1, 1], Parti[5, 4, 2, 1, 1, 1, 1, 1], Parti[5, 5, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8, 8}, {6, 5, 5}, {6, 6, 4}, {7, 5, 4}, {7, 6, 3}, {7, 7, 2}, {8, 4, 4}, \ {8, 5, 3}, {8, 6, 2}, {8, 7, 1}, {4, 4, 4, 4}, {5, 4, 4, 3}, {5, 5, 3, 3}, {5, 5, 4, \ 2}, {5, 5, 5, 1}, {6, 4, 3, 3}, {6, 4, 4, 2}, {6, 5, 3, 2}, {6, 5, 4, 1}, {6, \ 6, 2, 2}, {6, 6, 3, 1}, {7, 3, 3, 3}, {7, 4, 3, 2}, {7, 4, 4, 1}, {7, 5, 2, 2}, {7, \ 5, 3, 1}, {7, 6, 2, 1}, {7, 7, 1, 1}, {8, 3, 3, 2}, {8, 4, 2, 2}, {8, 4, 3, 1}, {8, \ 5, 2, 1}, {8, 6, 1, 1}, {4, 3, 3, 3, 3}, {4, 4, 3, 3, 2}, {4, 4, 4, 2, 2}, {4, 4, 4, \ 3, 1}, {5, 3, 3, 3, 2}, {5, 4, 3, 2, 2}, {5, 4, 3, 3, 1}, {5, 4, 4, 2, 1}, {5, 5, \ 2, 2, 2}, {5, 5, 3, 2, 1}, {5, 5, 4, 1, 1}, {6, 3, 3, 2, 2}, {6, 3, 3, 3, 1}, {6, 4, \ 2, 2, 2}, {6, 4, 3, 2, 1}, {6, 4, 4, 1, 1}, {6, 5, 2, 2, 1}, {6, 5, 3, 1, 1}, {6, 6, \ 2, 1, 1}, {7, 3, 2, 2, 2}, {7, 3, 3, 2, 1}, {7, 4, 2, 2, 1}, {7, 4, 3, 1, 1}, {7, 5, \ 2, 1, 1}, {7, 6, 1, 1, 1}, {8, 2, 2, 2, 2}, {8, 3, 2, 2, 1}, {8, 3, 3, 1, 1}, {8, 4, \ 2, 1, 1}, {8, 5, 1, 1, 1}, {3, 3, 3, 3, 2, 2}, {3, 3, 3, 3, 3, 1}, {4, 3, 3, 2, 2, \ 2}, {4, 3, 3, 3, 2, 1}, {4, 4, 2, 2, 2, 2}, {4, 4, 3, 2, 2, 1}, {4, 4, 3, 3, 1, \ 1}, {4, 4, 4, 2, 1, 1}, {5, 3, 2, 2, 2, 2}, {5, 3, 3, 2, 2, 1}, {5, 3, 3, 3, 1, \ 1}, {5, 4, 2, 2, 2, 1}, {5, 4, 3, 2, 1, 1}, {5, 4, 4, 1, 1, 1}, {5, 5, 2, 2, 1, \ 1}, {5, 5, 3, 1, 1, 1}, {6, 2, 2, 2, 2, 2}, {6, 3, 2, 2, 2, 1}, {6, 3, 3, 2, 1, \ 1}, {6, 4, 2, 2, 1, 1}, {6, 4, 3, 1, 1, 1}, {6, 5, 2, 1, 1, 1}, {6, 6, 1, 1, 1, \ 1}, {7, 2, 2, 2, 2, 1}, {7, 3, 2, 2, 1, 1}, {7, 3, 3, 1, 1, 1}, {7, 4, 2, 1, 1, \ 1}, {7, 5, 1, 1, 1, 1}, {3, 3, 2, 2, 2, 2, 2}, {3, 3, 3, 2, 2, 2, 1}, {3, 3, 3, 3, 2, 1, 1}, {4, 2, 2, 2, 2, 2, 2}, {4, 3, 2, 2, 2, 2, 1}, {4, 3, 3, 2, 2, 1, 1}, {4, 3, 3, 3, 1, 1, 1}, {4, 4, 2, 2, 2, 1, 1}, {4, 4, 3, 2, 1, 1, 1}, {4, 4, 4, 1, 1, 1, 1}, {5, 2, 2, 2, 2, 2, 1}, {5, 3, 2, 2, 2, 1, 1}, {5, 3, 3, 2, 1, 1, 1}, {5, 4, 2, 2, 1, 1, 1}, {5, 4, 3, 1, 1, 1, 1}, {5, 5, 2, 1, 1, 1, 1}, {6, 2, 2, 2, 2, 1, 1}, {6, 3, 2, 2, 1, 1, 1}, {6, 3, 3, 1, 1, 1, 1}, {6, 4, 2, 1, 1, 1, 1}, {6, 5, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 2}, {3, 2, 2, 2, 2, 2, 2, 1}, {3, 3, 2, 2, 2, 2, 1, 1}, {3, 3, 3, 2, 2, 1, 1, 1}, {3, 3, 3, 3, 1, 1, 1, \ 1}, {4, 2, 2, 2, 2, 2, 1, 1}, {4, 3, 2, 2, 2, 1, 1, 1}, {4, 3, 3, 2, 1, 1, 1, \ 1}, {4, 4, 2, 2, 1, 1, 1, 1}, {4, 4, 3, 1, 1, 1, 1, 1}, {5, 2, 2, 2, 2, 1, 1, \ 1}, {5, 3, 2, 2, 1, 1, 1, 1}, {5, 3, 3, 1, 1, 1, 1, 1}, {5, 4, 2, 1, 1, 1, 1, \ 1}, {5, 5, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 127\ \>", "\<\ 127\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part16,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[16], Parti[9, 7], Parti[10, 6], Parti[11, 5], Parti[12, 4], \ Parti[13, 3], Parti[14, 2], Parti[15, 1], Parti[9, 4, 3], Parti[9, 5, 2], Parti[9, 6, 1], \ Parti[10, 3, 3], Parti[10, 4, 2], Parti[10, 5, 1], Parti[11, 3, 2], \ Parti[11, 4, 1], Parti[12, 2, 2], Parti[12, 3, 1], Parti[13, 2, 1], Parti[14, 1, 1], \ Parti[9, 3, 2, 2], Parti[9, 3, 3, 1], Parti[9, 4, 2, 1], Parti[9, 5, 1, 1], Parti[10, 2, 2, \ 2], Parti[10, 3, 2, 1], Parti[10, 4, 1, 1], Parti[11, 2, 2, 1], Parti[11, 3, 1, \ 1], Parti[12, 2, 1, 1], Parti[13, 1, 1, 1], Parti[9, 2, 2, 2, 1], Parti[9, 3, \ 2, 1, 1], Parti[9, 4, 1, 1, 1], Parti[10, 2, 2, 1, 1], Parti[10, 3, 1, 1, 1], Parti[11, 2, 1, 1, 1], Parti[12, 1, 1, 1, 1], Parti[8, 2, 2, 2, 1, 1], Parti[8, 3, 2, 1, 1, 1], Parti[8, 4, 1, 1, 1, 1], Parti[9, 2, 2, 1, 1, 1], Parti[9, 3, 1, 1, 1, 1], Parti[10, 2, 1, 1, 1, 1], Parti[11, 1, 1, 1, 1, \ 1], Parti[7, 2, 2, 2, 1, 1, 1], Parti[7, 3, 2, 1, 1, 1, 1], Parti[7, 4, 1, 1, \ 1, 1, 1], Parti[8, 2, 2, 1, 1, 1, 1], Parti[8, 3, 1, 1, 1, 1, 1], Parti[9, 2, 1, 1, \ 1, 1, 1], Parti[10, 1, 1, 1, 1, 1, 1], Parti[6, 2, 2, 2, 1, 1, 1, 1], Parti[6, 3, 2, 1, 1, 1, 1, 1], Parti[6, 4, 1, 1, 1, 1, 1, 1], Parti[7, 2, 2, 1, 1, 1, 1, 1], Parti[7, 3, 1, 1, 1, 1, 1, 1], Parti[8, 2, 1, 1, 1, 1, 1, 1], Parti[9, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 1, 1], Parti[3, 2, 2, 2, 2, 2, 1, 1, 1], Parti[3, 3, 2, 2, 2, 1, 1, 1, 1], Parti[3, 3, 3, 2, 1, 1, 1, 1, 1], Parti[4, 2, 2, 2, 2, 1, 1, 1, 1], Parti[4, 3, 2, 2, 1, 1, 1, 1, 1], Parti[4, 3, 3, 1, 1, 1, 1, 1, 1], Parti[4, 4, 2, 1, 1, 1, 1, 1, 1], Parti[5, 2, 2, 2, 1, 1, 1, 1, 1], Parti[5, 3, 2, 1, 1, 1, 1, 1, 1], Parti[5, 4, 1, 1, 1, 1, 1, 1, 1], Parti[6, 2, 2, 1, 1, 1, 1, 1, 1], Parti[6, 3, 1, 1, 1, 1, 1, 1, 1], Parti[7, 2, 1, 1, 1, 1, 1, 1, 1], Parti[8, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 1, 1, 1, 1], Parti[3, 2, 2, 2, 2, 1, 1, 1, 1, 1], Parti[3, 3, 2, 2, 1, 1, 1, 1, 1, 1], Parti[3, 3, 3, 1, 1, 1, 1, 1, 1, 1], Parti[4, 2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[4, 3, 2, 1, 1, 1, 1, 1, 1, 1], Parti[4, 4, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 2, 2, 1, 1, 1, 1, 1, 1, 1], Parti[5, 3, 1, 1, 1, 1, 1, 1, 1, 1], Parti[6, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[7, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1], Parti[3, 2, 2, 2, 1, 1, 1, 1, 1, 1, \ 1], Parti[3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 2, 2, 1, 1, 1, 1, 1, 1, 1, \ 1], Parti[4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 2, 1, 1, 1, 1, 1, 1, 1, 1, \ 1], Parti[6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 2, 2, 1, 1, 1, 1, 1, 1, \ 1, 1], Parti[3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 3, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1], Parti[4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[5, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1], Parti[2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Parti[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{16}, {9, 7}, {10, 6}, {11, 5}, {12, 4}, {13, 3}, {14, 2}, {15, 1}, {9, 4, \ 3}, {9, 5, 2}, {9, 6, 1}, {10, 3, 3}, {10, 4, 2}, {10, 5, 1}, {11, 3, 2}, {11, \ 4, 1}, {12, 2, 2}, {12, 3, 1}, {13, 2, 1}, {14, 1, 1}, {9, 3, 2, 2}, {9, 3, 3, 1}, \ {9, 4, 2, 1}, {9, 5, 1, 1}, {10, 2, 2, 2}, {10, 3, 2, 1}, {10, 4, 1, 1}, {11, 2, 2, 1}, {11, 3, 1, 1}, {12, 2, 1, 1}, {13, 1, 1, 1}, {9, 2, 2, 2, \ 1}, {9, 3, 2, 1, 1}, {9, 4, 1, 1, 1}, {10, 2, 2, 1, 1}, {10, 3, 1, 1, 1}, {11, 2, 1, 1, 1}, {12, 1, 1, 1, 1}, {8, 2, 2, 2, 1, 1}, {8, 3, 2, 1, 1, 1}, \ {8, 4, 1, 1, 1, 1}, {9, 2, 2, 1, 1, 1}, {9, 3, 1, 1, 1, 1}, {10, 2, 1, 1, \ 1, 1}, {11, 1, 1, 1, 1, 1}, {7, 2, 2, 2, 1, 1, 1}, {7, 3, 2, 1, 1, 1, 1}, {7, 4, 1, 1, 1, 1, 1}, {8, 2, 2, 1, 1, 1, 1}, {8, 3, 1, 1, 1, 1, 1}, {9, 2, 1, 1, 1, 1, 1}, {10, 1, 1, 1, 1, 1, 1}, {6, 2, 2, 2, 1, 1, 1, 1}, {6, 3, 2, 1, 1, 1, 1, 1}, {6, 4, 1, 1, 1, 1, 1, 1}, {7, 2, 2, 1, 1, 1, 1, \ 1}, {7, 3, 1, 1, 1, 1, 1, 1}, {8, 2, 1, 1, 1, 1, 1, 1}, {9, 1, 1, 1, 1, 1, 1, \ 1}, {2, 2, 2, 2, 2, 2, 2, 1, 1}, {3, 2, 2, 2, 2, 2, 1, 1, 1}, {3, 3, 2, 2, 2, \ 1, 1, 1, 1}, {3, 3, 3, 2, 1, 1, 1, 1, 1}, {4, 2, 2, 2, 2, 1, 1, 1, 1}, {4, 3, 2, 2, 1, \ 1, 1, 1, 1}, {4, 3, 3, 1, 1, 1, 1, 1, 1}, {4, 4, 2, 1, 1, 1, 1, 1, 1}, {5, 2, 2, 2, 1, \ 1, 1, 1, 1}, {5, 3, 2, 1, 1, 1, 1, 1, 1}, {5, 4, 1, 1, 1, 1, 1, 1, 1}, {6, 2, 2, 1, 1, \ 1, 1, 1, 1}, {6, 3, 1, 1, 1, 1, 1, 1, 1}, {7, 2, 1, 1, 1, 1, 1, 1, 1}, {8, 1, 1, 1, 1, \ 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 1, 1, 1, 1}, {3, 2, 2, 2, 2, 1, 1, 1, 1, 1}, {3, 3, 2, 2, 1, 1, 1, 1, 1, 1}, {3, 3, 3, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 2, 2, 1, 1, 1, 1, 1, 1}, {4, 3, 2, 1, 1, 1, 1, 1, 1, 1}, {4, 4, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 2, 1, 1, 1, 1, 1, 1, 1}, {5, 3, 1, 1, 1, 1, 1, 1, 1, 1}, {6, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {7, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}, \ {3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, \ {5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1}, {3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 3432, 3640, 2548, 1260, 440, 104, 15, 69888, 65520, 24960, 21840, \ 38220, 20020, 13860, 10752, 2640, 3900, 896, 105, 114660, 112112, 180180, 71500, \ 23100, 71680, 43120, 14300, 16848, 4004, 455, 80640, 194040, 105600, 42120, 45760, \ 11648, 1365, 155232, 337920, 173250, 82368, 85800, 24024, 3003, 197120, 404250, \ 197120, 114400, 115830, 36608, 5005, 173250, 337920, 155232, 115830, 114400, 42042, \ 6435, 3432, 24960, 65520, 69888, 71500, 180180, 112112, 114660, 105600, 194040, \ 80640, 85800, 82368, 36608, 6435, 3640, 20020, 38220, 21840, 43120, 71680, 23100, \ 45760, 42120, 24024, 5005, 2548, 10752, 13860, 16848, 14300, 11648, 3003, 1260, \ 3900, 2640, 4004, 1365, 440, 896, 455, 104, 105, 15, 1]\ \>", "\<\ {1, 3432, 3640, 2548, 1260, 440, 104, 15, 69888, 65520, 24960, 21840, 38220, \ 20020, 13860, 10752, 2640, 3900, 896, 105, 114660, 112112, 180180, 71500, 23100, \ 71680, 43120, 14300, 16848, 4004, 455, 80640, 194040, 105600, 42120, 45760, 11648, \ 1365, 155232, 337920, 173250, 82368, 85800, 24024, 3003, 197120, 404250, 197120, \ 114400, 115830, 36608, 5005, 173250, 337920, 155232, 115830, 114400, 42042, 6435, \ 3432, 24960, 65520, 69888, 71500, 180180, 112112, 114660, 105600, 194040, 80640, 85800, \ 82368, 36608, 6435, 3640, 20020, 38220, 21840, 43120, 71680, 23100, 45760, 42120, \ 24024, 5005, 2548, 10752, 13860, 16848, 14300, 11648, 3003, 1260, 3900, 2640, \ 4004, 1365, 440, 896, 455, 104, 105, 15, 1}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["idealnumber = Plus @@ dims", "Input"], Cell[OutputFormData["\<\ 5954628\ \>", "\<\ 5954628\ \>"], "Output"] }, Open ]], Cell["Ad pleth22m4:", "Text"], Cell[CellGroupData[{ Cell["list = HoldList @@ pleth22m4", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8, 8], Parti[6, 6, 4], Parti[8, 4, 4], Parti[8, 6, 2], 2*Parti[4, 4, 4, 4], 3*Parti[5, 5, 3, 3], 3*Parti[6, 4, 4, 2], Parti[6, 5, \ 3, 2], Parti[6, 5, 4, 1], 3*Parti[6, 6, 2, 2], Parti[7, 4, 3, 2], 2*Parti[7, 5, 3, \ 1], Parti[7, 6, 2, 1], Parti[7, 7, 1, 1], Parti[8, 4, 2, 2], 3*Parti[4, 4, 4, \ 2, 2], 2*Parti[5, 3, 3, 3, 2], 2*Parti[5, 4, 3, 2, 2], 2*Parti[5, 4, 3, 3, 1], 2*Parti[5, 4, 4, 2, 1], 2*Parti[5, 5, 3, 2, 1], 2*Parti[5, 5, 4, 1, 1], Parti[6, 3, 3, 3, 1], 3*Parti[6, 4, 2, 2, 2], 3*Parti[6, 4, 3, 2, 1], 2*Parti[6, 5, 2, 2, 1], 2*Parti[6, 5, 3, 1, 1], Parti[7, 3, 3, 2, 1], Parti[7, 4, 2, 2, 1], Parti[7, 4, 3, 1, 1], Parti[7, 5, 2, 1, 1], Parti[8, 2, 2, 2, 2], Parti[3, 3, 3, 3, 2, 2], Parti[4, 3, 3, 3, 2, 1], 3*Parti[4, 4, 2, 2, 2, 2], Parti[4, 4, 3, 2, 2, 1], 3*Parti[4, 4, 3, 3, 1, \ 1], 2*Parti[5, 3, 3, 2, 2, 1], 2*Parti[5, 4, 2, 2, 2, 1], 3*Parti[5, 4, 3, 2, \ 1, 1], Parti[5, 4, 4, 1, 1, 1], 3*Parti[5, 5, 2, 2, 1, 1], Parti[6, 2, 2, 2, 2, \ 2], Parti[6, 3, 2, 2, 2, 1], 2*Parti[6, 3, 3, 2, 1, 1], 2*Parti[6, 4, 3, 1, 1, \ 1], Parti[6, 5, 2, 1, 1, 1], Parti[6, 6, 1, 1, 1, 1], Parti[7, 3, 2, 2, 1, 1], Parti[4, 2, 2, 2, 2, 2, 2], Parti[4, 3, 2, 2, 2, 2, 1], 2*Parti[4, 3, 3, 2, \ 2, 1, 1], Parti[4, 4, 3, 2, 1, 1, 1], Parti[5, 3, 2, 2, 2, 1, 1], Parti[5, 3, 3, 2, \ 1, 1, 1], Parti[5, 4, 2, 2, 1, 1, 1], Parti[5, 4, 3, 1, 1, 1, 1], Parti[6, 4, 2, 1, \ 1, 1, 1], Parti[2, 2, 2, 2, 2, 2, 2, 2], Parti[3, 3, 2, 2, 2, 2, 1, 1], Parti[3, 3, 3, 3, 1, 1, 1, 1], Parti[4, 4, 2, 2, 1, 1, 1, 1], Parti[5, 5, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8, 8}, {6, 6, 4}, {8, 4, 4}, {8, 6, 2}, 2 {4, 4, 4, 4}, 3 {5, 5, 3, 3}, 3 {6, 4, 4, 2}, {6, 5, 3, 2}, {6, 5, 4, 1}, 3 {6, 6, 2, 2}, {7, 4, 3, 2}, 2 {7, 5, 3, 1}, {7, 6, 2, 1}, {7, 7, 1, 1}, {8, 4, 2, 2}, 3 {4, 4, 4, 2, \ 2}, 2 {5, 3, 3, 3, 2}, 2 {5, 4, 3, 2, 2}, 2 {5, 4, 3, 3, 1}, 2 {5, 4, 4, 2, 1}, \ 2 {5, 5, 3, 2, 1}, 2 {5, 5, 4, 1, 1}, {6, 3, 3, 3, 1}, 3 {6, 4, 2, 2, 2}, 3 {6, 4, 3, 2, 1}, 2 {6, 5, 2, 2, 1}, 2 {6, 5, 3, 1, 1}, {7, 3, 3, 2, 1}, {7, 4, 2, 2, 1}, {7, 4, 3, 1, 1}, {7, 5, 2, 1, 1}, {8, 2, 2, 2, 2}, {3, 3, 3, 3, 2, 2}, {4, 3, 3, 3, 2, 1}, 3 {4, 4, 2, 2, 2, 2}, {4, 4, 3, 2, \ 2, 1}, 3 {4, 4, 3, 3, 1, 1}, 2 {5, 3, 3, 2, 2, 1}, 2 {5, 4, 2, 2, 2, 1}, 3 {5, 4, 3, 2, 1, 1}, {5, 4, 4, 1, 1, 1}, 3 {5, 5, 2, 2, 1, 1}, {6, 2, 2, \ 2, 2, 2}, {6, 3, 2, 2, 2, 1}, 2 {6, 3, 3, 2, 1, 1}, 2 {6, 4, 3, 1, 1, 1}, {6, 5, 2, \ 1, 1, 1}, {6, 6, 1, 1, 1, 1}, {7, 3, 2, 2, 1, 1}, {4, 2, 2, 2, 2, 2, 2}, {4, 3, 2, 2, \ 2, 2, 1}, 2 {4, 3, 3, 2, 2, 1, 1}, {4, 4, 3, 2, 1, 1, 1}, {5, 3, 2, 2, 2, 1, 1}, {5, 3, 3, 2, 1, 1, 1}, {5, 4, 2, 2, 1, 1, 1}, {5, 4, 3, 1, 1, 1, 1}, {6, 4, 2, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 2}, {3, 3, 2, 2, 2, 2, 1, 1}, {3, 3, 3, 3, 1, 1, 1, 1}, {4, 4, 2, 2, 1, 1, 1, 1}, {5, 5, 1, 1, 1, 1, 1, \ 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 63\ \>", "\<\ 63\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["list = list /. _*x_Parti :> x", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[8, 8], Parti[6, 6, 4], Parti[8, 4, 4], Parti[8, 6, 2], \ Parti[4, 4, 4, 4], Parti[5, 5, 3, 3], Parti[6, 4, 4, 2], Parti[6, 5, 3, 2], Parti[6, 5, 4, 1], \ Parti[6, 6, 2, 2], Parti[7, 4, 3, 2], Parti[7, 5, 3, 1], Parti[7, 6, 2, 1], \ Parti[7, 7, 1, 1], Parti[8, 4, 2, 2], Parti[4, 4, 4, 2, 2], Parti[5, 3, 3, \ 3, 2], Parti[5, 4, 3, 2, 2], Parti[5, 4, 3, 3, 1], Parti[5, 4, 4, 2, 1], Parti[5, 5, 3, 2, 1], Parti[5, 5, 4, 1, 1], Parti[6, 3, 3, 3, 1], Parti[6, 4, 2, 2, 2], Parti[6, 4, 3, 2, 1], Parti[6, 5, 2, 2, 1], Parti[6, 5, 3, 1, 1], Parti[7, 3, 3, 2, 1], Parti[7, 4, 2, 2, 1], Parti[7, 4, 3, 1, 1], Parti[7, 5, 2, 1, 1], Parti[8, 2, 2, 2, 2], Parti[3, 3, 3, 3, 2, 2], Parti[4, 3, 3, 3, 2, 1], Parti[4, 4, 2, 2, 2, 2], Parti[4, 4, 3, 2, 2, 1], Parti[4, 4, 3, 3, 1, 1], Parti[5, 3, 3, 2, 2, 1], Parti[5, 4, 2, 2, 2, 1], Parti[5, 4, 3, 2, 1, 1], Parti[5, 4, 4, 1, 1, 1], Parti[5, 5, 2, 2, 1, 1], Parti[6, 2, 2, 2, 2, 2], Parti[6, 3, 2, 2, 2, 1], Parti[6, 3, 3, 2, 1, 1], Parti[6, 4, 3, 1, 1, 1], Parti[6, 5, 2, 1, 1, 1], Parti[6, 6, 1, 1, 1, 1], Parti[7, 3, 2, 2, 1, 1], Parti[4, 2, 2, 2, 2, 2, \ 2], Parti[4, 3, 2, 2, 2, 2, 1], Parti[4, 3, 3, 2, 2, 1, 1], Parti[4, 4, 3, 2, \ 1, 1, 1], Parti[5, 3, 2, 2, 2, 1, 1], Parti[5, 3, 3, 2, 1, 1, 1], Parti[5, 4, 2, 2, \ 1, 1, 1], Parti[5, 4, 3, 1, 1, 1, 1], Parti[6, 4, 2, 1, 1, 1, 1], Parti[2, 2, 2, 2, \ 2, 2, 2, 2], Parti[3, 3, 2, 2, 2, 2, 1, 1], Parti[3, 3, 3, 3, 1, 1, 1, 1], Parti[4, 4, 2, 2, 1, 1, 1, 1], Parti[5, 5, 1, 1, 1, 1, 1, 1]]\ \>", "\<\ {{8, 8}, {6, 6, 4}, {8, 4, 4}, {8, 6, 2}, {4, 4, 4, 4}, {5, 5, 3, 3}, {6, 4, \ 4, 2}, {6, 5, 3, 2}, {6, 5, 4, 1}, {6, 6, 2, 2}, {7, 4, 3, 2}, {7, 5, 3, 1}, {7, \ 6, 2, 1}, {7, 7, 1, 1}, {8, 4, 2, 2}, {4, 4, 4, 2, 2}, {5, 3, 3, 3, 2}, {5, 4, 3, 2, \ 2}, {5, 4, 3, 3, 1}, {5, 4, 4, 2, 1}, {5, 5, 3, 2, 1}, {5, 5, 4, 1, 1}, {6, 3, \ 3, 3, 1}, {6, 4, 2, 2, 2}, {6, 4, 3, 2, 1}, {6, 5, 2, 2, 1}, {6, 5, 3, 1, 1}, {7, 3, \ 3, 2, 1}, {7, 4, 2, 2, 1}, {7, 4, 3, 1, 1}, {7, 5, 2, 1, 1}, {8, 2, 2, 2, 2}, {3, 3, 3, 3, 2, 2}, {4, 3, 3, 3, 2, 1}, {4, 4, 2, 2, 2, 2}, {4, 4, 3, 2, 2, \ 1}, {4, 4, 3, 3, 1, 1}, {5, 3, 3, 2, 2, 1}, {5, 4, 2, 2, 2, 1}, {5, 4, 3, 2, 1, \ 1}, {5, 4, 4, 1, 1, 1}, {5, 5, 2, 2, 1, 1}, {6, 2, 2, 2, 2, 2}, {6, 3, 2, 2, 2, \ 1}, {6, 3, 3, 2, 1, 1}, {6, 4, 3, 1, 1, 1}, {6, 5, 2, 1, 1, 1}, {6, 6, 1, 1, 1, \ 1}, {7, 3, 2, 2, 1, 1}, {4, 2, 2, 2, 2, 2, 2}, {4, 3, 2, 2, 2, 2, 1}, {4, 3, 3, 2, 2, 1, 1}, {4, 4, 3, 2, 1, 1, 1}, {5, 3, 2, 2, 2, 1, 1}, {5, 3, 3, 2, 1, 1, 1}, {5, 4, 2, 2, 1, 1, 1}, {5, 4, 3, 1, 1, 1, 1}, {6, 4, 2, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2, 2, 2}, {3, 3, 2, 2, 2, 2, 1, 1}, {3, 3, 3, 3, 1, 1, 1, 1}, {4, 4, 2, 2, 1, 1, 1, 1}, {5, 5, 1, 1, 1, 1, 1, \ 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Length[list]", "Input"], Cell[OutputFormData["\<\ 63\ \>", "\<\ 63\ \>"], "Output"] }, Open ]], Cell["\<\ Dimensions of the minimal left ideals belonging to these partitions:\ \>", "Text"], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ list", "Input"], Cell[OutputFormData["\<\ HoldList[1430, 51480, 60060, 68640, 24024, 171600, 336336, 500500, 292864, \ 150150, 512512, 416988, 210210, 32032, 262080, 171600, 280280, 640640, 549120, \ 549120, 640640, 280280, 360360, 480480, 1153152, 630630, 648648, 582400, 698880, 673920, \ 515970, 76440, 51480, 292864, 150150, 500500, 336336, 648648, 630630, 1153152, \ 360360, 480480, 91728, 559104, 819000, 819000, 559104, 91728, 600600, 32032, 210210, \ 416988, 512512, 515970, 673920, 698880, 582400, 600600, 1430, 68640, 60060, 262080, \ 76440]\ \>", "\<\ {1430, 51480, 60060, 68640, 24024, 171600, 336336, 500500, 292864, 150150, \ 512512, 416988, 210210, 32032, 262080, 171600, 280280, 640640, 549120, 549120, \ 640640, 280280, 360360, 480480, 1153152, 630630, 648648, 582400, 698880, 673920, 515970, \ 76440, 51480, 292864, 150150, 500500, 336336, 648648, 630630, 1153152, 360360, 480480, \ 91728, 559104, 819000, 819000, 559104, 91728, 600600, 32032, 210210, 416988, \ 512512, 515970, 673920, 698880, 582400, 600600, 1430, 68640, 60060, 262080, 76440}\ \>"], "Output"] }, Open ]], Cell[TextData[ "We determine the smallest and the largest dimension of an minimal left ideal \ within the decomposition of [2,2] \[CircleDot] [4]."], "Text"], Cell[CellGroupData[{ Cell["Min @@ dims", "Input"], Cell[OutputFormData["\<\ 1430\ \>", "\<\ 1430\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Max @@ dims", "Input"], Cell[OutputFormData["\<\ 1153152\ \>", "\<\ 1153152\ \>"], "Output"] }, Open ]], Cell[TextData[ "Further we calculate the memory (in MByte) which a 1153152 \[Times] \ 1153152-matrix filled with 2-Byte-Integers needs."], "Text"], Cell[CellGroupData[{ Cell["%*%*2 Byte", "Input"], Cell[OutputFormData["\<\ 2659519070208*Byte\ \>", "\<\ 2659519070208 Byte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 kByte/Byte)", "Input"], Cell[OutputFormData["\<\ 2597186592*kByte\ \>", "\<\ 2597186592 kByte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["%*(1/1024 MByte/kByte)", "Input"], Cell[OutputFormData["\<\ (81162081*MByte)/32\ \>", "\<\ 81162081 MByte -------------- 32\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["N[%]", "Input"], Cell[OutputFormData["\<\ 2.53631503125*^6*MByte\ \>", "\<\ 6 2.53632 10 MByte\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["restparts = Complement[part16,list]", "Input"], Cell[OutputFormData["\<\ HoldList[Parti[16], Parti[9, 7], Parti[10, 6], Parti[11, 5], Parti[12, 4], \ Parti[13, 3], Parti[14, 2], Parti[15, 1], Parti[6, 5, 5], Parti[7, 5, 4], Parti[7, 6, 3], \ Parti[7, 7, 2], Parti[8, 5, 3], Parti[8, 7, 1], Parti[9, 4, 3], Parti[9, 5, \ 2], Parti[9, 6, 1], Parti[10, 3, 3], Parti[10, 4, 2], Parti[10, 5, 1], \ Parti[11, 3, 2], Parti[11, 4, 1], Parti[12, 2, 2], Parti[12, 3, 1], Parti[13, 2, 1], \ Parti[14, 1, 1], Parti[5, 4, 4, 3], Parti[5, 5, 4, 2], Parti[5, 5, 5, 1], Parti[6, 4, 3, 3], \ Parti[6, 6, 3, 1], Parti[7, 3, 3, 3], Parti[7, 4, 4, 1], Parti[7, 5, 2, 2], \ Parti[8, 3, 3, 2], Parti[8, 4, 3, 1], Parti[8, 5, 2, 1], Parti[8, 6, 1, 1], \ Parti[9, 3, 2, 2], Parti[9, 3, 3, 1], Parti[9, 4, 2, 1], Parti[9, 5, 1, 1], \ Parti[10, 2, 2, 2], Parti[10, 3, 2, 1], Parti[10, 4, 1, 1], Parti[11, 2, 2, \ 1], Parti[11, 3, 1, 1], Parti[12, 2, 1, 1], Parti[13, 1, 1, 1], Parti[4, 3, 3, \ 3, 3], Parti[4, 4, 3, 3, 2], Parti[4, 4, 4, 3, 1], Parti[5, 5, 2, 2, 2], Parti[6, 3, 3, 2, 2], Parti[6, 4, 4, 1, 1], Parti[6, 6, 2, 1, 1], Parti[7, 3, 2, 2, 2], Parti[7, 6, 1, 1, 1], Parti[8, 3, 2, 2, 1], Parti[8, 3, 3, 1, 1], Parti[8, 4, 2, 1, 1], Parti[8, 5, 1, 1, 1], Parti[9, 2, 2, 2, 1], Parti[9, 3, 2, 1, 1], Parti[9, 4, 1, 1, 1], Parti[10, 2, 2, 1, 1], Parti[10, 3, 1, 1, 1], Parti[11, 2, 1, 1, 1], Parti[12, 1, 1, 1, 1], Parti[3, 3, 3, 3, 3, 1], Parti[4, 3, 3, 2, 2, 2], Parti[4, 4, 4, 2, 1, 1], Parti[5, 3, 2, 2, 2, 2], Parti[5, 3, 3, 3, 1, 1], Parti[5, 5, 3, 1, 1, 1], Parti[6, 4, 2, 2, 1, 1], Parti[7, 2, 2, 2, 2, 1], Parti[7, 3, 3, 1, 1, 1], Parti[7, 4, 2, 1, 1, 1], Parti[7, 5, 1, 1, 1, 1], Parti[8, 2, 2, 2, 1, 1], Parti[8, 3, 2, 1, 1, 1], Parti[8, 4, 1, 1, 1, 1], Parti[9, 2, 2, 1, 1, 1], Parti[9, 3, 1, 1, 1, 1], Parti[10, 2, 1, 1, 1, 1], \ Parti[11, 1, 1, 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2, 1, 1, 1, 1, 1, 1, 1, 1}, {4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, \ {4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, \ {2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1}, {4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ 1, 1, 1}, {3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["dims = PartDim[#]& /@ restparts", "Input"], Cell[OutputFormData["\<\ HoldList[1, 3432, 3640, 2548, 1260, 440, 104, 15, 36036, 100100, 91520, \ 30030, 112112, 18018, 69888, 65520, 24960, 21840, 38220, 20020, 13860, 10752, 2640, 3900, \ 896, 105, 180180, 231660, 60060, 300300, 200200, 140140, 240240, 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1]\ \>", "\<\ {1, 3432, 3640, 2548, 1260, 440, 104, 15, 36036, 100100, 91520, 30030, \ 112112, 18018, 69888, 65520, 24960, 21840, 38220, 20020, 13860, 10752, 2640, 3900, 896, \ 105, 180180, 231660, 60060, 300300, 200200, 140140, 240240, 320320, 206388, 318500, \ 266240, 73710, 114660, 112112, 180180, 71500, 23100, 71680, 43120, 14300, 16848, 4004, \ 455, 60060, 231660, 180180, 250250, 448448, 411840, 240240, 305760, 116480, 360360, \ 280280, 429000, 150150, 80640, 194040, 105600, 42120, 45760, 11648, 1365, 36036, \ 200200, 300300, 240240, 411840, 448448, 928746, 168168, 429000, 630630, 200200, \ 155232, 337920, 173250, 82368, 85800, 24024, 3003, 30030, 91520, 100100, 240240, \ 320320, 140140, 116480, 305760, 200200, 630630, 429000, 168168, 197120, 404250, \ 197120, 114400, 115830, 36608, 5005, 18018, 112112, 73710, 266240, 318500, 206388, \ 150150, 429000, 280280, 360360, 173250, 337920, 155232, 115830, 114400, 42042, \ 6435, 3432, 24960, 65520, 69888, 71500, 180180, 112112, 114660, 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