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(*CacheID: 232*)


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(*NotebookOptionsPosition[    932907,      14641]*)
(*NotebookOutlinePosition[    933589,      14665]*)
(*  CellTagsIndexPosition[    933545,      14661]*)
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Notebook[{

Cell[CellGroupData[{
Cell["Dodecaphonic Matrices", "Title"],

Cell["\<\
Periodic Matrices of period up to 12, pseudo-antisymmetric ones of the \
(-1,0,1)-variety 
(Mathematicians object to calling this 'signed binary', but I like that \
description,
and 'pseudo-antisymmetric' should be 'sum of an antisymmetric and a diagonal \
matrix').\
\>", "Subsubtitle"],

Cell[BoxData[
    \(see : \ \(\(http\)\(:\)\) // 
      mathworld . wolfram . com/PeriodicMatrix . html\)], "Input"],

Cell[CellGroupData[{

Cell["inits", "Subsection"],

Cell[BoxData[
    \(triamat[li_List]\  := 
      Block[{len = Sqrt[8  Length[li] + 1]/2 - 1/2}, 
        If[IntegerQ[len], \(Part[li, #] &\) /@ 
            Table[If[j > i, j \((j - 1)\)/2\  + i, 
                i \((i - 1)\)/2\  + j], {i, len}, {j, len}], li]]\)], "Input",\

  InitializationCell->True,
  Background->RGBColor[1, 1, 0]],

Cell[BoxData[
    \(triamatsig[li_List]\  := 
      Block[{len = Sqrt[8  Length[li] + 1]/2 - 1/2}, 
        If[IntegerQ[
            len], \((\(Part[li, #] &\) /@ 
                Table[If[j > i, j \((j - 1)\)/2\  + i, 
                    i \((i - 1)\)/2\  + j], {i, len}, {j, len}])\) 
            Table[If[j > i, \(-1\), 1], {i, len}, {j, len}], li]]\)], "Input",\

  InitializationCell->True,
  Background->RGBColor[1, 1, 0]],

Cell[BoxData[
    \(belongs[m_?MatrixQ] := 
      Block[{ident = IdentityMatrix[Length[m]]}, 
        NestWhile[m\  . \ # &, 
            m, \((\((# =!= 
                      ident)\) && \((Union[
                        Flatten[{#, {\(-1\), 0, 1}}]] === {\(-1\), 0, 
                        1})\) && \((Det[#] =!= 0)\))\) &]\  === 
          ident]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(powerlist[m_?MatrixQ] := 
      Block[{ident = IdentityMatrix[Length[m]]}, 
        NestWhileList[m\  . \ # &, 
          m, \((\((# =!= 
                    ident)\) && \((Union[
                      Flatten[{#, {\(-1\), 0, 1}}]] === {\(-1\), 0, 
                      1})\) && \((Det[#] =!= 0)\))\) &]\ ]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(powerlength[\((q_?MatrixQ)\)] := 
      Length@NestWhileList[q . \ # &, q, # =!= IdentityMatrix[Length[q]] &, 
          1, 13]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(pseudoantisymmetric[
        q_?MatrixQ] := \((q + Transpose[q] === 
          2\ DiagonalMatrix[
              MapIndexed[Part[#1, First[#2]] &, q, 1]])\)\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(pseudoantisymmetric[
        q_?MatrixQ] := \((q + Transpose[q])\) \((1 - 
              IdentityMatrix[Length[q]])\) === 0\ q\)], "Input",
  InitializationCell->True],

Cell["\<\
ordering[li_List]:=Last/@Sort[Transpose[{li,Range[Length[li]]}]]\
\>", "Input",
  InitializationCell->True],

Cell["not really happy with this: max 11 or max 12 is different !!", "Text"],

Cell[BoxData[
    \(sortedform[li_List] := 
      FixedPoint[
        Part[Transpose[Sort[Transpose[#]]]\ , \ ordering[Transpose[#]]\ ] &, 
        li, 11]\)], "Input",
  InitializationCell->True],

Cell["check it:", "Text"],

Cell[CellGroupData[{

Cell["\<\
{belongs[#],powerlength[#],pseudoantisymmetric[#]}& @ IdentityMatrix[5]\
\>", "Input"],

Cell[BoxData[
    \({True, 1, True}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
the cases n= 1 to 6 {2,14,92,796,7672, 83944,...?...} A072148\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["n=1", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(n = 1; 
    it = triamatsig /@ \((\(-1\) + 
            IntegerDigits[Range[0, \(-1\) + 3^\((n \((n + 1)\)/2)\)], 3, 
              n \((n + 1)\)/2])\);\), "\[IndentingNewLine]", 
    \(result1 = 
      Cases[it, \((q_?MatrixQ)\) /; \ 
          Det[q] =!= 0\  && 
            And @@ Table[\[IndentingNewLine]Union[
                    Flatten[{MatrixPower[\ \ q\ , k], {\(-1\), 0, 
                          1}}]] === {\(-1\), 0, 1}, \[IndentingNewLine]{k, 
                  12}]\ ]\)}], "Input"],

Cell[BoxData[
    \({{{\(-1\)}}, {{1}}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[
          Function[{a}, 
              Length@NestWhileList[a . \ # &, a, # =!= IdentityMatrix[1] &]] /@ 
            result1]]\)], "Input"],

Cell[BoxData[
    \({{1, 1}, {1, 2}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
            result1]]\)], "Input"],

Cell[BoxData[
    \({{2, 1}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(Det[#] &\)\  /@ \ result1]]\)], "Input"],

Cell[BoxData[
    \({{1, \(-1\)}, {1, 1}}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["n=2", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(n = 2; 
    it = triamatsig /@ \((\(-1\) + 
            IntegerDigits[Range[0, \(-1\) + 3^\((n \((n + 1)\)/2)\)], 3, 
              n \((n + 1)\)/2])\);\), "\[IndentingNewLine]", 
    \(result2 = 
      Cases[it, \((q_?MatrixQ)\) /; \ 
          Det[q] =!= 0\  && 
            And @@ Table[\[IndentingNewLine]Union[
                    Flatten[{MatrixPower[\ \ q\ , k], {\(-1\), 0, 
                          1}}]] === {\(-1\), 0, 1}, \[IndentingNewLine]{k, 
                  12}]\ ]\)}], "Input"],

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          1}, {\(-1\), \(-1\)}}, {{0, 1}, {\(-1\), 0}}, {{0, 1}, {\(-1\), 
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          0}}}\)], "Output"]
}, Closed]],

Cell["\<\
Map[MatrixForm,NestList[#.(result2[[ 2 ]])&,IdentityMatrix[2],14],{1}]\
\>", "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[result2]\)], "Input"],

Cell[BoxData[
    \(14\)], "Output"]
}, Open  ]],

Cell["frequency of powerlengths", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\ powerlength /@ result2]]\)], "Input"],

Cell[BoxData[
    \({{1, 1}, {3, 2}, {4, 3}, {2, 4}, {4, 6}}\)], "Output"]
}, Open  ]],

Cell[" frequency of count non-zero elements", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
            result2]]\)], "Input"],

Cell[BoxData[
    \({{6, 2}, {8, 3}}\)], "Output"]
}, Open  ]],

Cell["frequency of characteristic polys", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\ \(CoefficientList[CharacteristicPolynomial[#, x], 
                x] &\)\  /@ \ result2\ ]]\)], "Input"],

Cell[BoxData[
    \({{2, {\(-1\), 0, 1}}, {1, {1, \(-2\), 1}}, {4, {1, \(-1\), 1}}, {2, {1, 
          0, 1}}, {4, {1, 1, 1}}, {1, {1, 2, 1}}}\)], "Output"]
}, Open  ]],

Cell["frequency of Det", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[Det\  /@ \ result2]]\)], "Input"],

Cell[BoxData[
    \({{2, \(-1\)}, {12, 1}}\)], "Output"]
}, Open  ]],

Cell["All resulting matrices are pseudo-antisymmetric,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result3, \((q_?
            MatrixQ)\) /; \((\(! pseudoantisymmetric[q]\))\)]\)], "Input"],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Union[pseudoantisymmetric /@ result2]\)], "Input"],

Cell[BoxData[
    \({True}\)], "Output"]
}, Open  ]],

Cell["\<\
but their powers need not all be pseudo-antisymmetric !  they are for 2x2 \
matices though.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result2, \((q_?MatrixQ)\) /; \ 
        Not[And @@ \((pseudoantisymmetric /@ \ 
                NestWhileList[q\  . \ # &, 
                  q, # =!= IdentityMatrix[Length[q]] &]\ )\)]]\)], "Input"],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Position[
      Function[{a}, \ 
          NestWhileList[a . \ # &, a, # =!= IdentityMatrix[2] &]] /@ 
        result2, \((q_?
            MatrixQ)\) /; \((\(! pseudoantisymmetric[q]\))\)]\)], "Input"],

Cell[BoxData[
    \({}\)], "Output"]
}, Open  ]],

Cell["\<\
sorted form :  one set  of  4 matrices with equivalent sorted forms, 4 sets \
of 2 equivalents,  2 sets of 1 equivalent.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ \((\ \ Split[
          Sort[\ \ Length /@ 
              Split[Sort[\ \ \ sortedform\ \  /@ \ 
                    result2]]\ \ ]]\ )\)\)], "Input"],

Cell[BoxData[
    \({{2, 1}, {6, 2}}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["n=3", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(n = 3; 
    it = triamatsig /@ \((\(-1\) + 
            IntegerDigits[Range[0, \(-1\) + 3^\((n \((n + 1)\)/2)\)], 3, 
              n \((n + 1)\)/2])\);\), "\[IndentingNewLine]", 
    \(result3 = 
      Cases[it, \((q_?MatrixQ)\) /; \ 
          Det[q] =!= 0\  && 
            And @@ Table[\[IndentingNewLine]Union[
                    Flatten[{MatrixPower[\ \ q\ , k], {\(-1\), 0, 
                          1}}]] === {\(-1\), 0, 1}, \[IndentingNewLine]{k, 
                  12}]\ ]\)}], "Input"],

Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[result3]\)], "Input"],

Cell[BoxData[
    \(92\)], "Output"]
}, Open  ]],

Cell["do all powers themselves belong?", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result3, \((q_?MatrixQ)\) /; 
        Union[belongs /@ powerlist[q]] =!= {True}]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell["\<\
do the powers themselves produce extra (-1,0,1)-matrices whose powers are \
same, but non-pseudoantisymmetric ones?\
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result3, \((q_?MatrixQ)\) /; 
        Union[pseudoantisymmetric /@ powerlist[q]] =!= {True}]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(24\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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        pseudoantisymmetric[#] === False &]\)], "Input",
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          1}, {\(-1\), 0, \(-1\)}, {\(-1\), \(-1\), 0}}}\)], "Output"]
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Cell[" so these are all (-1,0,1)-matrices whose powers are same:", "Text",
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          1}, {1, \(-1\), \(-1\)}}, {{0, \(-1\), \(-1\)}, {1, \(-1\), \
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          1}, {1, \(-1\), 1}, {1, \(-1\), 0}}, {{0, \(-1\), 1}, {1, 
          1, \(-1\)}, {1, 1, 0}}, {{0, 0, \(-1\)}, {0, \(-1\), 0}, {\(-1\), 
          0, 0}}, {{0, 0, 1}, {0, \(-1\), 0}, {1, 0, 0}}, {{0, 
          1, \(-1\)}, {\(-1\), \(-1\), 1}, {\(-1\), \(-1\), 0}}, {{0, 
          1, \(-1\)}, {\(-1\), 1, \(-1\)}, {\(-1\), 1, 0}}, {{0, 
          1, \(-1\)}, {1, 0, \(-1\)}, {1, 1, \(-1\)}}, {{0, 1, \(-1\)}, {1, 
          0, 1}, {1, \(-1\), 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 
          0, \(-1\)}}, {{0, 1, 1}, {\(-1\), \(-1\), \(-1\)}, {1, 1, 0}}, {{0, 
          1, 1}, {\(-1\), 1, 1}, {1, \(-1\), 0}}, {{0, 1, 1}, {1, 
          0, \(-1\)}, {\(-1\), 1, 1}}, {{0, 1, 1}, {1, 0, 
          1}, {\(-1\), \(-1\), \(-1\)}}, {{1, \(-1\), \(-1\)}, {1, 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["frequency of powerlengths", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength /@ result3]]\)], "Input"],

Cell[BoxData[
    \({{1, 1}, {7, 2}, {12, 3}, {36, 4}, {36, 6}}\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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      Split[Sort[powerlength /@ all3]]\)], "Input"],

Cell[BoxData[
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Cell[" frequency of count non-zero elements", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["All resulting matrices are pseudo-antisymmetric,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
but their powers need not all be pseudo-antisymmetric !  24 matrices of the \
total 92  have at least one 'funny' power:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \(24\)], "Output"]
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Cell["the  funny ones are:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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          Not[And @@ \((pseudoantisymmetric /@ \ powerlist[q])\)]]\)], "Input"],

Cell[BoxData[
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          1, 1}, {\(-1\), \(-1\), 0}, {\(-1\), 0, \(-1\)}}, {{1, 1, 
          0}, {\(-1\), \(-1\), 1}, {0, \(-1\), 1}}, {{1, 1, 
          0}, {\(-1\), \(-1\), \(-1\)}, {0, 1, 1}}, {{1, 
          1, \(-1\)}, {\(-1\), \(-1\), 0}, {1, 0, \(-1\)}}, {{1, 0, 1}, {0, 
          1, 1}, {\(-1\), \(-1\), \(-1\)}}, {{1, 0, 1}, {0, 
          1, \(-1\)}, {\(-1\), 1, \(-1\)}}, {{1, 0, \(-1\)}, {0, 1, 
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          1, \(-1\)}}, {{1, \(-1\), 1}, {1, \(-1\), 0}, {\(-1\), 
          0, \(-1\)}}, {{1, \(-1\), 0}, {1, \(-1\), 1}, {0, \(-1\), 
          1}}, {{1, \(-1\), 0}, {1, \(-1\), \(-1\)}, {0, 1, 
          1}}, {{1, \(-1\), \(-1\)}, {1, \(-1\), 0}, {1, 
          0, \(-1\)}}}\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["these are fully pseudoantisymmetric over all their powers:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 
      57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 
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}, Open  ]],

Cell["\<\
so the funny powers are found as power 2 and 3 of a selection of 24  of the \
36 matrices having powerlength 4.\
\>", "Text"],

Cell["Aha, there also are 24 matrices with 7 non-zero elements:", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
and, Aha again, all funny ones have 7 non-zero elements, and vice versa : all \
those with 7 non-zero elements are funny:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
sorted form :  two sets of  9 matrices with equivalent sorted forms, 4 sets \
of 6 equivalents,  8 sets of 3 equivalents ...\
\>", "Text"],

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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
sorted form :  two sets of  9 matrices with equivalent sorted forms, 4 sets \
of 6 equivalents,  8 sets of 3 equivalents ...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell["n=4", "Subsubsection"],

Cell[BoxData[{
    \(n = 4; 
    it = triamatsig /@ \((\(-1\) + 
            IntegerDigits[Range[0, \(-1\) + 3^\((n \((n + 1)\)/2)\)], 3, 
              n \((n + 1)\)/2])\);\), "\[IndentingNewLine]", 
    \(result4 = 
      Cases[it, \((q_?MatrixQ)\) /; \ 
          Det[q] =!= 0\  && 
            And @@ Table[\[IndentingNewLine]Union[
                    Flatten[{MatrixPower[\ \ q\ , k], {\(-1\), 0, 
                          1}}]] === {\(-1\), 0, 1}, \[IndentingNewLine]{k, 
                  12}]\ ]\)}], "Input",
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  FontColor->GrayLevel[0.666667]],

Cell[BoxData[
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"\<H:/Data/Notebooks/pseudoAntisymmMatrixPowers_4.txt\>";\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell["do all powers themselves belong?", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
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      result4, \((q_?MatrixQ)\) /; 
        Union[belongs /@ powerlist[q]] =!= {True}]\)], "Input",
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Cell[BoxData[
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Cell["\<\
do the powers themselves produce extra (-1,0,1)-matrices whose powers are \
same, but non-pseudoantisymmetric ones?\
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
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        Union[pseudoantisymmetric /@ powerlist[q]] =!= {True}]\)], "Input",
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[" so these are all (-1,0,1)-matrices whose powers are same:", "Text",
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          0, 0, \(-1\)}, {0, 0, \(-1\), 0}, {1, \(-1\), 0, 0}}, {{1, \(-1\), 
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          0}}, {{1, \(-1\), 0, 1}, {1, 0, 0, 1}, {0, 0, \(-1\), 0}, {\(-1\), 
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          1, 0, 0}}, {{1, \(-1\), 1, 0}, {1, 0, 1, 0}, {\(-1\), 1, 0, 0}, {0, 
          0, 0, \(-1\)}}, {{1, \(-1\), 1, 0}, {1, 0, 1, 0}, {\(-1\), 1, 0, 
          0}, {0, 0, 0, 1}}, {{1, 0, \(-1\), \(-1\)}, {0, \(-1\), 0, 0}, {1, 
          0, 0, \(-1\)}, {1, 0, \(-1\), 0}}, {{1, 0, \(-1\), \(-1\)}, {0, 1, 
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          0, \(-1\), 1}, {0, 1, 0, 0}, {1, 0, 0, 1}, {\(-1\), 0, 1, 0}}, {{1, 
          0, 0, 0}, {0, \(-1\), \(-1\), \(-1\)}, {0, 1, 0, 1}, {0, 1, 1, 
          0}}, {{1, 0, 0, 0}, {0, \(-1\), \(-1\), 1}, {0, 1, 
          0, \(-1\)}, {0, \(-1\), \(-1\), 0}}, {{1, 0, 0, 0}, {0, \(-1\), 0, 
          0}, {0, 0, 0, \(-1\)}, {0, 0, \(-1\), 0}}, {{1, 0, 0, 
          0}, {0, \(-1\), 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}}, {{1, 0, 0, 
          0}, {0, \(-1\), 1, \(-1\)}, {0, \(-1\), 0, \(-1\)}, {0, 1, \(-1\), 
          0}}, {{1, 0, 0, 0}, {0, \(-1\), 1, 1}, {0, \(-1\), 0, 
          1}, {0, \(-1\), 1, 0}}, {{1, 0, 0, 0}, {0, 
          0, \(-1\), \(-1\)}, {0, \(-1\), 0, \(-1\)}, {0, 1, 1, 1}}, {{1, 0, 
          0, 0}, {0, 0, \(-1\), \(-1\)}, {0, \(-1\), 0, 1}, {0, 
          1, \(-1\), \(-1\)}}, {{1, 0, 0, 0}, {0, 0, \(-1\), \(-1\)}, {0, 
          1, \(-1\), \(-1\)}, {0, \(-1\), 1, 0}}, {{1, 0, 0, 0}, {0, 
          0, \(-1\), \(-1\)}, {0, 1, 1, 1}, {0, \(-1\), \(-1\), 0}}, {{1, 0, 
          0, 0}, {0, 0, \(-1\), 0}, {0, \(-1\), 0, 0}, {0, 0, 
          0, \(-1\)}}, {{1, 0, 0, 0}, {0, 0, \(-1\), 1}, {0, \(-1\), 
          0, \(-1\)}, {0, \(-1\), 1, \(-1\)}}, {{1, 0, 0, 0}, {0, 0, \(-1\), 
          1}, {0, \(-1\), 0, 1}, {0, \(-1\), \(-1\), 1}}, {{1, 0, 0, 0}, {0, 
          0, \(-1\), 1}, {0, 1, \(-1\), 1}, {0, 1, \(-1\), 0}}, {{1, 0, 0, 
          0}, {0, 0, \(-1\), 1}, {0, 1, 1, \(-1\)}, {0, 1, 1, 0}}, {{1, 0, 0, 
          0}, {0, 0, 0, \(-1\)}, {0, 0, \(-1\), 0}, {0, \(-1\), 0, 0}}, {{1, 
          0, 0, 0}, {0, 0, 0, 1}, {0, 0, \(-1\), 0}, {0, 1, 0, 0}}, {{1, 0, 
          0, 0}, {0, 0, 1, \(-1\)}, {0, \(-1\), \(-1\), 
          1}, {0, \(-1\), \(-1\), 0}}, {{1, 0, 0, 0}, {0, 0, 
          1, \(-1\)}, {0, \(-1\), 1, \(-1\)}, {0, \(-1\), 1, 0}}, {{1, 0, 0, 
          0}, {0, 0, 1, \(-1\)}, {0, 1, 0, \(-1\)}, {0, 1, 1, \(-1\)}}, {{1, 
          0, 0, 0}, {0, 0, 1, \(-1\)}, {0, 1, 0, 1}, {0, 1, \(-1\), 1}}, {{1, 
          0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}, {0, 0, 0, \(-1\)}}, {{1, 0, 
          0, 0}, {0, 0, 1, 1}, {0, \(-1\), \(-1\), \(-1\)}, {0, 1, 1, 
          0}}, {{1, 0, 0, 0}, {0, 0, 1, 1}, {0, \(-1\), 1, 1}, {0, 1, \(-1\), 
          0}}, {{1, 0, 0, 0}, {0, 0, 1, 1}, {0, 1, 0, \(-1\)}, {0, \(-1\), 1, 
          1}}, {{1, 0, 0, 0}, {0, 0, 1, 1}, {0, 1, 0, 
          1}, {0, \(-1\), \(-1\), \(-1\)}}, {{1, 0, 0, 0}, {0, 
          1, \(-1\), \(-1\)}, {0, 1, 0, \(-1\)}, {0, 1, \(-1\), 0}}, {{1, 0, 
          0, 0}, {0, 1, \(-1\), 1}, {0, 1, 0, 1}, {0, \(-1\), 1, 0}}, {{1, 0, 
          0, 0}, {0, 1, 1, \(-1\)}, {0, \(-1\), 0, 1}, {0, 1, 1, 0}}, {{1, 0, 
          0, 0}, {0, 1, 1, 1}, {0, \(-1\), 0, \(-1\)}, {0, \(-1\), \(-1\), 
          0}}, {{1, 0, 1, \(-1\)}, {0, \(-1\), 0, 0}, {\(-1\), 0, 0, 1}, {1, 
          0, 1, 0}}, {{1, 0, 1, \(-1\)}, {0, 1, 0, 0}, {\(-1\), 0, 0, 1}, {1, 
          0, 1, 0}}, {{1, 0, 1, 1}, {0, \(-1\), 0, 0}, {\(-1\), 0, 
          0, \(-1\)}, {\(-1\), 0, \(-1\), 0}}, {{1, 0, 1, 1}, {0, 1, 0, 
          0}, {\(-1\), 0, 0, \(-1\)}, {\(-1\), 0, \(-1\), 0}}, {{1, 
          1, \(-1\), 0}, {\(-1\), 0, 1, 0}, {1, 1, 0, 0}, {0, 0, 
          0, \(-1\)}}, {{1, 1, \(-1\), 0}, {\(-1\), 0, 1, 0}, {1, 1, 0, 
          0}, {0, 0, 0, 1}}, {{1, 1, 0, \(-1\)}, {\(-1\), 0, 0, 1}, {0, 
          0, \(-1\), 0}, {1, 1, 0, 0}}, {{1, 1, 0, \(-1\)}, {\(-1\), 0, 0, 
          1}, {0, 0, 1, 0}, {1, 1, 0, 0}}, {{1, 1, 0, 1}, {\(-1\), 0, 
          0, \(-1\)}, {0, 0, \(-1\), 0}, {\(-1\), \(-1\), 0, 0}}, {{1, 1, 0, 
          1}, {\(-1\), 0, 0, \(-1\)}, {0, 0, 1, 0}, {\(-1\), \(-1\), 0, 
          0}}, {{1, 1, 1, 0}, {\(-1\), 0, \(-1\), 0}, {\(-1\), \(-1\), 0, 
          0}, {0, 0, 0, \(-1\)}}, {{1, 1, 1, 0}, {\(-1\), 0, \(-1\), 
          0}, {\(-1\), \(-1\), 0, 0}, {0, 0, 0, 1}}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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        w[{\(-1\), 2, 0, \(-2\), 1}, 2]}, {1, 
        w[{1, \(-4\), 6, \(-4\), 1}, 4]}, {24, 
        w[{1, \(-3\), 4, \(-3\), 1}, 3]}, {108, 
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        w[{1, \(-2\), 3, \(-2\), 1}, 2]}, {24, 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["All resulting matrices are pseudo-antisymmetric,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
but their powers need not all be pseudo-antisymmetric !  192 matrices of the \
total 796  have at least one 'funny' power:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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        Not[And @@ \((pseudoantisymmetric /@ \ 
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Cell[BoxData[
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Cell[BoxData[
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Cell["the indices of the funny ones are:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \({16, 17, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 36, 37, 40, 
      41, 47, 48, 49, 50, 52, 53, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 75, 
      77, 78, 79, 81, 82, 89, 91, 98, 100, 105, 106, 107, 108, 110, 111, 117, 
      118, 119, 120, 122, 123, 177, 178, 179, 180, 181, 182, 183, 184, 189, 
      191, 198, 200, 208, 210, 215, 217, 219, 220, 221, 222, 231, 232, 233, 
      234, 250, 251, 254, 255, 257, 258, 259, 260, 261, 262, 263, 264, 266, 
      267, 270, 271, 526, 527, 530, 531, 533, 534, 535, 536, 537, 538, 539, 
      540, 542, 543, 546, 547, 563, 564, 565, 566, 575, 576, 577, 578, 580, 
      582, 587, 589, 597, 599, 606, 608, 613, 614, 615, 616, 617, 618, 619, 
      620, 674, 675, 677, 678, 679, 680, 686, 687, 689, 690, 691, 692, 697, 
      699, 706, 708, 715, 716, 718, 719, 720, 722, 727, 729, 730, 731, 733, 
      734, 735, 736, 737, 738, 744, 745, 747, 748, 749, 750, 756, 757, 760, 
      761, 764, 765, 767, 768, 769, 770, 771, 772, 773, 774, 776, 777, 780, 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["these are fully pseudoantisymmetric over all their powers:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 22, 31, 34, 
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      73, 74, 76, 80, 83, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 99, 
      101, 102, 103, 104, 109, 112, 113, 114, 115, 116, 121, 124, 125, 126, 
      127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 
      141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 
      155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 
      169, 170, 171, 172, 173, 174, 175, 176, 185, 186, 187, 188, 190, 192, 
      193, 194, 195, 196, 197, 199, 201, 202, 203, 204, 205, 206, 207, 209, 
      211, 212, 213, 214, 216, 218, 223, 224, 225, 226, 227, 228, 229, 230, 
      235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 
      249, 252, 253, 256, 265, 268, 269, 272, 273, 274, 275, 276, 277, 278, 
      279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 
      293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 
      307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 
      321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 
      335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 
      349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 
      363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 
      377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 
      391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 
      405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 
      419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 
      433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 
      447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 
      461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 
      475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 
      489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 
      503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 
      517, 518, 519, 520, 521, 522, 523, 524, 525, 528, 529, 532, 541, 544, 
      545, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 
      561, 562, 567, 568, 569, 570, 571, 572, 573, 574, 579, 581, 583, 584, 
      585, 586, 588, 590, 591, 592, 593, 594, 595, 596, 598, 600, 601, 602, 
      603, 604, 605, 607, 609, 610, 611, 612, 621, 622, 623, 624, 625, 626, 
      627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 
      641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 
      655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 
      669, 670, 671, 672, 673, 676, 681, 682, 683, 684, 685, 688, 693, 694, 
      695, 696, 698, 700, 701, 702, 703, 704, 705, 707, 709, 710, 711, 712, 
      713, 714, 717, 721, 723, 724, 725, 726, 728, 732, 739, 740, 741, 742, 
      743, 746, 751, 752, 753, 754, 755, 758, 759, 762, 763, 766, 775, 778, 
      779, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 
      795, 796}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ result4]]\)], "Input"],

Cell[BoxData[
    \({{1, 1}, {15, 2}, {72, 3}, {300, 4}, {312, 6}, {96, 12}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ \((result4[\([funny]\)])\)]]\)], "Input"],

Cell[BoxData[
    \({{192, 4}}\)], "Output"]
}, Closed]],

Cell["\<\
so the funny powers are found as power 2 and 3 of a selection of 192  of the \
300 matrices having powerlength 4.\
\>", "Text"],

Cell["\<\
Aha, there also are 192 matrices with 6 or 8 non-zero elements:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
            result4]]\)], "Input"],

Cell[BoxData[
    \({{76, 4}, {288, 5}, {192, 6}, {192, 8}, {48, 12}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Flatten[#])\) &\)\  /@ \ result4]]\)], "Input"],

Cell[BoxData[
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}, Closed]],

Cell["\<\
and, Aha again, all funny ones have 8 non-zero elements, and vice versa : all \
those with 8 non-zero elements are funny:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ \ \ \ Abs[Flatten[#]]\ \ \ )\) &\)\  /@ \ 
            result4[\([funny]\)]]]\)], "Input"],

Cell[BoxData[
    \({{192, 8}}\)], "Output"]
}, Closed]],

Cell["\<\
sorted form :  one set of  42 matrices with equivalent sorted forms, 2 sets \
of 32 equivalents,  4 sets of 24 equivalents ...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ \((\ \ Split[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell["n=5", "Subsubsection"],

Cell[BoxData[
    \(Table[n = 5; \n
      it = triamatsig /@ \((\(-1\) + 
              IntegerDigits[Range[k\ \ 3^10, \(-1\) + \((k + 1)\)\ 3^10], 3, 
                n \((n + 1)\)/2])\); \[IndentingNewLine]Cases[
        it, \((q_?MatrixQ)\) /; \ 
          Det[q] =!= 0\  && 
            And @@ Table[\[IndentingNewLine]Union[
                    Flatten[{MatrixPower[\ \ q\ , j], {\(-1\), 0, 
                          1}}]] === {\(-1\), 0, 1}, \[IndentingNewLine]{j, 
                  n^2}]\ ]\[IndentingNewLine], {k, 
        0, \(-1\) + 3^5}]\)], "Input",
  Evaluatable->False,
  FontColor->GrayLevel[0.666667]],

Cell[BoxData[
    \(\(result5 = \ << \
"\<H:/Data/Notebooks/pseudoAntisymmMatrixPowers_5.txt\>";\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[result5]\)], "Input"],

Cell[BoxData[
    \(7672\)], "Output"]
}, Open  ]],

Cell["do all powers themselves belong?", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result5, \((q_?MatrixQ)\) /; 
        Union[belongs /@ powerlist[q]] =!= {True}]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(0\)], "Output"]
}, Closed]],

Cell["\<\
do the powers themselves produce extra (-1,0,1)-matrices whose powers are \
same, but non-pseudoantisymmetric ones?\
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result5, \((q_?MatrixQ)\) /; 
        Union[pseudoantisymmetric /@ powerlist[q]] =!= {True}]\)], "Input",
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Cell[BoxData[
    \(3360\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(powers5 = 
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Cell[TextData[{
  "Would it be sufficient to show that for some A and B,  \
CharacteristicEqn(A)=CharacteristicEqn(B) \nbut Tr(A) <> Tr(B) to illustrate \
that De Rham's conjecture is false? ",
  ButtonBox["De Rham",
    ButtonData:>{
      URL[ "http://nyjm.albany.edu:8000/~mark/derham.html"], None},
    ButtonStyle->"Hyperlink"],
  "  (<A HREF=http://nyjm.albany.edu:8000/~mark/derham.html> De Rham </A> )"
}], "Text"],

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Cell["All resulting matrices are pseudo-antisymmetric,", "Text"],

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Cell["\<\
but their powers need not all be pseudo-antisymmetric !  3360 matrices of the \
total 7672  have at least one 'funny' power:\
\>", "Text"],

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Cell["the indices of the funny ones are:", "Text"],

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      1195, 1196, 1197, 1198, 1199, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 
      1207, 1208, 1209, 1210, 1211, 1212, 1213, 1214, 1215, 1216, 1217, 1218, 
      1219, 1220, 1221, 1222, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1230, 
      1231, 1232, 1233, 1234, 1235, 1236, 1237, 1238, 1239, 1240, 1243, 1244, 
      1245, 1246, 1247, 1249, 1254, 1256, 1271, 1272, 1273, 1274, 1277, 1278, 
      1279, 1280, 1281, 1283, 1288, 1290, 1305, 1306, 1307, 1308, 1311, 1312, 
      1313, 1314, 1315, 1317, 1322, 1324, 1339, 1340, 1341, 1342, 1343, 1344, 
      1345, 1346, 1347, 1348, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1363, 
      1368, 1370, 1371, 1373, 1378, 1380, 1381, 1383, 1388, 1390, 1415, 1417, 
      1422, 1424, 1425, 1427, 1432, 1434, 1435, 1437, 1442, 1444, 1445, 1446, 
      1447, 1448, 1449, 1450, 1457, 1458, 1459, 1460, 1461, 1462, 1609, 1610, 
      1611, 1612, 1613, 1615, 1620, 1622, 1637, 1638, 1639, 1640, 1643, 1644, 
      1645, 1646, 1647, 1649, 1654, 1656, 1671, 1672, 1673, 1674, 1677, 1678, 
      1679, 1680, 1681, 1683, 1688, 1690, 1705, 1706, 1707, 1708, 1733, 1734, 
      1735, 1736, 1737, 1738, 1739, 1740, 1741, 1742, 1743, 1744, 1745, 1746, 
      1747, 1748, 1749, 1750, 1751, 1752, 1753, 1754, 1755, 1756, 1757, 1758, 
      1759, 1760, 1761, 1762, 1763, 1764, 1765, 1766, 1767, 1768, 1769, 1770, 
      1771, 1772, 1773, 1774, 1775, 1776, 1777, 1778, 1779, 1780, 1781, 1782, 
      1783, 1784, 1799, 1801, 1806, 1808, 1809, 1810, 1811, 1812, 1815, 1817, 
      1818, 1820, 1821, 1822, 1823, 1824, 1826, 1827, 1828, 1829, 1831, 1832, 
      1833, 1834, 1835, 1836, 1837, 1838, 1840, 1841, 1842, 1843, 1845, 1846, 
      1847, 1848, 1849, 1851, 1852, 1854, 1855, 1856, 1857, 1858, 1859, 1861, 
      1862, 1864, 1865, 1867, 1868, 1870, 1871, 1872, 1873, 1874, 1875, 1876, 
      1877, 1878, 1879, 1881, 1882, 1884, 1885, 1887, 1888, 1890, 1891, 1892, 
      1893, 1894, 1897, 1898, 1899, 1900, 1901, 1903, 1908, 1910, 1925, 1926, 
      1927, 1928, 1930, 1931, 1932, 1933, 1941, 1942, 1943, 1944, 1946, 1947, 
      1948, 1949, 1951, 1952, 1953, 1954, 1955, 1957, 1962, 1964, 1971, 1972, 
      1973, 1974, 1975, 1976, 1978, 1979, 1981, 1983, 1988, 1990, 1992, 1993, 
      1995, 1996, 2003, 2005, 2006, 2008, 2015, 2017, 2018, 2020, 2027, 2028, 
      2029, 2030, 2031, 2032, 2033, 2034, 2035, 2037, 2042, 2044, 2046, 2047, 
      2048, 2049, 2057, 2058, 2059, 2060, 2062, 2063, 2064, 2065, 2139, 2141, 
      2146, 2148, 2149, 2150, 2151, 2152, 2153, 2155, 2156, 2158, 2159, 2160, 
      2161, 2162, 2169, 2171, 2172, 2174, 2175, 2176, 2177, 2178, 2185, 2187, 
      2188, 2190, 2197, 2199, 2200, 2202, 2210, 2211, 2213, 2214, 2215, 2217, 
      2222, 2224, 2225, 2226, 2228, 2229, 2231, 2232, 2233, 2234, 2241, 2243, 
      2244, 2246, 2247, 2248, 2249, 2250, 2257, 2259, 2260, 2262, 2263, 2265, 
      2270, 2272, 2273, 2274, 2275, 2276, 2277, 2278, 2279, 2280, 2281, 2283, 
      2284, 2286, 2287, 2289, 2290, 2292, 2293, 2294, 2295, 2296, 2297, 2298, 
      2299, 2300, 2301, 2303, 2304, 2306, 2307, 2309, 2310, 2312, 2313, 2314, 
      2315, 2316, 2319, 2320, 2321, 2322, 2323, 2325, 2330, 2332, 2347, 2348, 
      2349, 2350, 2351, 2352, 2353, 2354, 2369, 2371, 2376, 2378, 2379, 2380, 
      2381, 2382, 2385, 2387, 2388, 2390, 2391, 2392, 2393, 2394, 2396, 2397, 
      2398, 2399, 2401, 2402, 2403, 2404, 2405, 2406, 2407, 2408, 2410, 2411, 
      2412, 2413, 2415, 2416, 2417, 2418, 2419, 2421, 2422, 2424, 2427, 2428, 
      2429, 2430, 2431, 2433, 2438, 2440, 2455, 2456, 2457, 2458, 2483, 2484, 
      2485, 2486, 2501, 2503, 2508, 2510, 2511, 2512, 2513, 2514, 2520, 2521, 
      2522, 2523, 2525, 2526, 2527, 2528, 2529, 2530, 2532, 2533, 2534, 2535, 
      2542, 2543, 2544, 2545, 2547, 2548, 2549, 2550, 2551, 2552, 2554, 2555, 
      2556, 2557, 2562, 2563, 2564, 2565, 2568, 2569, 2570, 2571, 2573, 2574, 
      2575, 2576, 2577, 2578, 2579, 2580, 2581, 2582, 2583, 2584, 2585, 2586, 
      2587, 2588, 2590, 2591, 2592, 2593, 2596, 2597, 2598, 2599, 2601, 2602, 
      2603, 2604, 2605, 2606, 2607, 2608, 2609, 2610, 2611, 2612, 2613, 2614, 
      2615, 2616, 2617, 2618, 2619, 2620, 2621, 2622, 2623, 2624, 2625, 2626, 
      2627, 2628, 2629, 2630, 2631, 2632, 2633, 2634, 2635, 2636, 2637, 2638, 
      2639, 2640, 2641, 2642, 2643, 2644, 2645, 2646, 2647, 2648, 2650, 2651, 
      2652, 2653, 2656, 2657, 2658, 2659, 2661, 2662, 2663, 2664, 2665, 2666, 
      2667, 2668, 2669, 2670, 2671, 2672, 2673, 2674, 2675, 2676, 2678, 2679, 
      2680, 2681, 2684, 2685, 2686, 2687, 2692, 2693, 2694, 2695, 2697, 2698, 
      2699, 2700, 2701, 2702, 2704, 2705, 2706, 2707, 2714, 2715, 2716, 2717, 
      2719, 2720, 2721, 2722, 2723, 2724, 2726, 2727, 2728, 2729, 2735, 2736, 
      2737, 2738, 2739, 2741, 2746, 2748, 2763, 2764, 2765, 2766, 2791, 2792, 
      2793, 2794, 2809, 2811, 2816, 2818, 2819, 2820, 2821, 2822, 2827, 2828, 
      2829, 2830, 2831, 2833, 2838, 2840, 2855, 2856, 2857, 2858, 2883, 2884, 
      2885, 2886, 2901, 2903, 2908, 2910, 2911, 2912, 2913, 2914, 2919, 2920, 
      2921, 2922, 2923, 2925, 2930, 2932, 2947, 2948, 2949, 2950, 2975, 2976, 
      2977, 2978, 2993, 2995, 3000, 3002, 3003, 3004, 3005, 3006, 3011, 3012, 
      3013, 3014, 3015, 3017, 3022, 3024, 3039, 3040, 3041, 3042, 3045, 3046, 
      3047, 3048, 3049, 3051, 3056, 3058, 3073, 3074, 3075, 3076, 3079, 3080, 
      3081, 3082, 3083, 3085, 3090, 3092, 3107, 3108, 3109, 3110, 3135, 3136, 
      3137, 3138, 3139, 3140, 3141, 3142, 3143, 3144, 3145, 3146, 3147, 3148, 
      3149, 3150, 3151, 3152, 3153, 3154, 3155, 3156, 3157, 3158, 3159, 3160, 
      3161, 3162, 3163, 3164, 3165, 3166, 3167, 3168, 3169, 3170, 3171, 3172, 
      3173, 3174, 3175, 3176, 3177, 3178, 3179, 3180, 3181, 3182, 3183, 3185, 
      3190, 3192, 3193, 3195, 3200, 3202, 3203, 3205, 3210, 3212, 3213, 3214, 
      3215, 3216, 3217, 3218, 3225, 3226, 3227, 3228, 3229, 3230, 3255, 3256, 
      3257, 3258, 3259, 3260, 3267, 3268, 3269, 3270, 3271, 3272, 3273, 3275, 
      3280, 3282, 3283, 3285, 3290, 3292, 3293, 3295, 3300, 3302, 3471, 3472, 
      3473, 3474, 3475, 3476, 3477, 3478, 3479, 3480, 3481, 3482, 3483, 3484, 
      3485, 3486, 3487, 3488, 3489, 3490, 3491, 3492, 3493, 3494, 3495, 3496, 
      3497, 3498, 3499, 3500, 3501, 3502, 3503, 3504, 3505, 3506, 3507, 3508, 
      3509, 3510, 3511, 3512, 3513, 3514, 3515, 3516, 3517, 3518, 3521, 3522, 
      3523, 3524, 3525, 3527, 3532, 3534, 3549, 3550, 3551, 3552, 3555, 3556, 
      3557, 3558, 3559, 3561, 3566, 3568, 3583, 3584, 3585, 3586, 3589, 3590, 
      3591, 3592, 3593, 3595, 3600, 3602, 3617, 3618, 3619, 3620, 4053, 4054, 
      4055, 4056, 4071, 4073, 4078, 4080, 4081, 4082, 4083, 4084, 4087, 4088, 
      4089, 4090, 4105, 4107, 4112, 4114, 4115, 4116, 4117, 4118, 4121, 4122, 
      4123, 4124, 4139, 4141, 4146, 4148, 4149, 4150, 4151, 4152, 4155, 4156, 
      4157, 4158, 4159, 4160, 4161, 4162, 4163, 4164, 4165, 4166, 4167, 4168, 
      4169, 4170, 4171, 4172, 4173, 4174, 4175, 4176, 4177, 4178, 4179, 4180, 
      4181, 4182, 4183, 4184, 4185, 4186, 4187, 4188, 4189, 4190, 4191, 4192, 
      4193, 4194, 4195, 4196, 4197, 4198, 4199, 4200, 4201, 4202, 4371, 4373, 
      4378, 4380, 4381, 4383, 4388, 4390, 4391, 4393, 4398, 4400, 4401, 4402, 
      4403, 4404, 4405, 4406, 4413, 4414, 4415, 4416, 4417, 4418, 4443, 4444, 
      4445, 4446, 4447, 4448, 4455, 4456, 4457, 4458, 4459, 4460, 4461, 4463, 
      4468, 4470, 4471, 4473, 4478, 4480, 4481, 4483, 4488, 4490, 4491, 4492, 
      4493, 4494, 4495, 4496, 4497, 4498, 4499, 4500, 4501, 4502, 4503, 4504, 
      4505, 4506, 4507, 4508, 4509, 4510, 4511, 4512, 4513, 4514, 4515, 4516, 
      4517, 4518, 4519, 4520, 4521, 4522, 4523, 4524, 4525, 4526, 4527, 4528, 
      4529, 4530, 4531, 4532, 4533, 4534, 4535, 4536, 4537, 4538, 4563, 4564, 
      4565, 4566, 4581, 4583, 4588, 4590, 4591, 4592, 4593, 4594, 4597, 4598, 
      4599, 4600, 4615, 4617, 4622, 4624, 4625, 4626, 4627, 4628, 4631, 4632, 
      4633, 4634, 4649, 4651, 4656, 4658, 4659, 4660, 4661, 4662, 4667, 4668, 
      4669, 4670, 4671, 4673, 4678, 4680, 4695, 4696, 4697, 4698, 4723, 4724, 
      4725, 4726, 4741, 4743, 4748, 4750, 4751, 4752, 4753, 4754, 4759, 4760, 
      4761, 4762, 4763, 4765, 4770, 4772, 4787, 4788, 4789, 4790, 4815, 4816, 
      4817, 4818, 4833, 4835, 4840, 4842, 4843, 4844, 4845, 4846, 4851, 4852, 
      4853, 4854, 4855, 4857, 4862, 4864, 4879, 4880, 4881, 4882, 4907, 4908, 
      4909, 4910, 4925, 4927, 4932, 4934, 4935, 4936, 4937, 4938, 4944, 4945, 
      4946, 4947, 4949, 4950, 4951, 4952, 4953, 4954, 4956, 4957, 4958, 4959, 
      4966, 4967, 4968, 4969, 4971, 4972, 4973, 4974, 4975, 4976, 4978, 4979, 
      4980, 4981, 4986, 4987, 4988, 4989, 4992, 4993, 4994, 4995, 4997, 4998, 
      4999, 5000, 5001, 5002, 5003, 5004, 5005, 5006, 5007, 5008, 5009, 5010, 
      5011, 5012, 5014, 5015, 5016, 5017, 5020, 5021, 5022, 5023, 5025, 5026, 
      5027, 5028, 5029, 5030, 5031, 5032, 5033, 5034, 5035, 5036, 5037, 5038, 
      5039, 5040, 5041, 5042, 5043, 5044, 5045, 5046, 5047, 5048, 5049, 5050, 
      5051, 5052, 5053, 5054, 5055, 5056, 5057, 5058, 5059, 5060, 5061, 5062, 
      5063, 5064, 5065, 5066, 5067, 5068, 5069, 5070, 5071, 5072, 5074, 5075, 
      5076, 5077, 5080, 5081, 5082, 5083, 5085, 5086, 5087, 5088, 5089, 5090, 
      5091, 5092, 5093, 5094, 5095, 5096, 5097, 5098, 5099, 5100, 5102, 5103, 
      5104, 5105, 5108, 5109, 5110, 5111, 5116, 5117, 5118, 5119, 5121, 5122, 
      5123, 5124, 5125, 5126, 5128, 5129, 5130, 5131, 5138, 5139, 5140, 5141, 
      5143, 5144, 5145, 5146, 5147, 5148, 5150, 5151, 5152, 5153, 5159, 5160, 
      5161, 5162, 5163, 5165, 5170, 5172, 5187, 5188, 5189, 5190, 5215, 5216, 
      5217, 5218, 5233, 5235, 5240, 5242, 5243, 5244, 5245, 5246, 5249, 5251, 
      5252, 5254, 5255, 5256, 5257, 5258, 5260, 5261, 5262, 5263, 5265, 5266, 
      5267, 5268, 5269, 5270, 5271, 5272, 5274, 5275, 5276, 5277, 5279, 5280, 
      5281, 5282, 5283, 5285, 5286, 5288, 5291, 5292, 5293, 5294, 5295, 5297, 
      5302, 5304, 5319, 5320, 5321, 5322, 5323, 5324, 5325, 5326, 5341, 5343, 
      5348, 5350, 5351, 5352, 5353, 5354, 5357, 5358, 5359, 5360, 5361, 5363, 
      5364, 5366, 5367, 5369, 5370, 5372, 5373, 5374, 5375, 5376, 5377, 5378, 
      5379, 5380, 5381, 5383, 5384, 5386, 5387, 5389, 5390, 5392, 5393, 5394, 
      5395, 5396, 5397, 5398, 5399, 5400, 5401, 5403, 5408, 5410, 5411, 5413, 
      5414, 5416, 5423, 5424, 5425, 5426, 5427, 5429, 5430, 5432, 5439, 5440, 
      5441, 5442, 5444, 5445, 5447, 5448, 5449, 5451, 5456, 5458, 5459, 5460, 
      5462, 5463, 5471, 5473, 5474, 5476, 5483, 5485, 5486, 5488, 5495, 5496, 
      5497, 5498, 5499, 5501, 5502, 5504, 5511, 5512, 5513, 5514, 5515, 5517, 
      5518, 5520, 5521, 5522, 5523, 5524, 5525, 5527, 5532, 5534, 5608, 5609, 
      5610, 5611, 5613, 5614, 5615, 5616, 5624, 5625, 5626, 5627, 5629, 5631, 
      5636, 5638, 5639, 5640, 5641, 5642, 5643, 5644, 5645, 5646, 5653, 5655, 
      5656, 5658, 5665, 5667, 5668, 5670, 5677, 5678, 5680, 5681, 5683, 5685, 
      5690, 5692, 5694, 5695, 5697, 5698, 5699, 5700, 5701, 5702, 5709, 5711, 
      5716, 5718, 5719, 5720, 5721, 5722, 5724, 5725, 5726, 5727, 5729, 5730, 
      5731, 5732, 5740, 5741, 5742, 5743, 5745, 5746, 5747, 5748, 5763, 5765, 
      5770, 5772, 5773, 5774, 5775, 5776, 5779, 5780, 5781, 5782, 5783, 5785, 
      5786, 5788, 5789, 5791, 5792, 5794, 5795, 5796, 5797, 5798, 5799, 5800, 
      5801, 5802, 5803, 5805, 5806, 5808, 5809, 5811, 5812, 5814, 5815, 5816, 
      5817, 5818, 5819, 5821, 5822, 5824, 5825, 5826, 5827, 5828, 5830, 5831, 
      5832, 5833, 5835, 5836, 5837, 5838, 5839, 5840, 5841, 5842, 5844, 5845, 
      5846, 5847, 5849, 5850, 5851, 5852, 5853, 5855, 5856, 5858, 5861, 5862, 
      5863, 5864, 5865, 5867, 5872, 5874, 5889, 5890, 5891, 5892, 5893, 5894, 
      5895, 5896, 5897, 5898, 5899, 5900, 5901, 5902, 5903, 5904, 5905, 5906, 
      5907, 5908, 5909, 5910, 5911, 5912, 5913, 5914, 5915, 5916, 5917, 5918, 
      5919, 5920, 5921, 5922, 5923, 5924, 5925, 5926, 5927, 5928, 5929, 5930, 
      5931, 5932, 5933, 5934, 5935, 5936, 5937, 5938, 5939, 5940, 5965, 5966, 
      5967, 5968, 5983, 5985, 5990, 5992, 5993, 5994, 5995, 5996, 5999, 6000, 
      6001, 6002, 6017, 6019, 6024, 6026, 6027, 6028, 6029, 6030, 6033, 6034, 
      6035, 6036, 6051, 6053, 6058, 6060, 6061, 6062, 6063, 6064, 6211, 6212, 
      6213, 6214, 6215, 6216, 6223, 6224, 6225, 6226, 6227, 6228, 6229, 6231, 
      6236, 6238, 6239, 6241, 6246, 6248, 6249, 6251, 6256, 6258, 6283, 6285, 
      6290, 6292, 6293, 6295, 6300, 6302, 6303, 6305, 6310, 6312, 6313, 6314, 
      6315, 6316, 6317, 6318, 6325, 6326, 6327, 6328, 6329, 6330, 6331, 6332, 
      6333, 6334, 6349, 6351, 6356, 6358, 6359, 6360, 6361, 6362, 6365, 6366, 
      6367, 6368, 6383, 6385, 6390, 6392, 6393, 6394, 6395, 6396, 6399, 6400, 
      6401, 6402, 6417, 6419, 6424, 6426, 6427, 6428, 6429, 6430, 6433, 6434, 
      6435, 6436, 6437, 6438, 6439, 6440, 6441, 6442, 6443, 6444, 6445, 6446, 
      6447, 6448, 6449, 6450, 6451, 6452, 6453, 6454, 6455, 6456, 6457, 6458, 
      6459, 6460, 6461, 6462, 6463, 6464, 6465, 6466, 6467, 6468, 6469, 6470, 
      6471, 6472, 6473, 6474, 6475, 6476, 6477, 6478, 6479, 6480, 6508, 6509, 
      6510, 6511, 6513, 6514, 6515, 6516, 6517, 6518, 6520, 6521, 6522, 6523, 
      6527, 6529, 6530, 6532, 6533, 6534, 6535, 6536, 6538, 6539, 6540, 6541, 
      6543, 6544, 6545, 6546, 6547, 6548, 6549, 6550, 6552, 6553, 6554, 6555, 
      6557, 6558, 6559, 6560, 6561, 6563, 6564, 6566, 6567, 6568, 6569, 6570, 
      6585, 6587, 6592, 6594, 6595, 6596, 6597, 6598, 6604, 6605, 6606, 6607, 
      6609, 6610, 6611, 6612, 6613, 6614, 6616, 6617, 6618, 6619, 6623, 6624, 
      6625, 6626, 6627, 6629, 6630, 6632, 6633, 6635, 6636, 6638, 6639, 6640, 
      6641, 6642, 6643, 6644, 6645, 6646, 6647, 6649, 6650, 6652, 6653, 6655, 
      6656, 6658, 6659, 6660, 6661, 6662, 6663, 6664, 6665, 6666, 6681, 6683, 
      6688, 6690, 6691, 6692, 6693, 6694, 6698, 6699, 6700, 6701, 6703, 6704, 
      6705, 6706, 6714, 6715, 6716, 6717, 6719, 6720, 6721, 6722, 6729, 6731, 
      6736, 6738, 6739, 6740, 6741, 6742, 6743, 6745, 6747, 6749, 6754, 6756, 
      6758, 6760, 6767, 6768, 6770, 6771, 6772, 6774, 6781, 6783, 6784, 6785, 
      6787, 6788, 6789, 6791, 6796, 6798, 6799, 6800, 6801, 6802, 6803, 6804, 
      6805, 6806, 6814, 6815, 6816, 6817, 6819, 6820, 6821, 6822, 6830, 6831, 
      6832, 6833, 6835, 6836, 6837, 6838, 6839, 6840, 6847, 6848, 6849, 6850, 
      6851, 6852, 6859, 6861, 6866, 6868, 6869, 6871, 6876, 6878, 6879, 6881, 
      6886, 6888, 6901, 6903, 6908, 6910, 6911, 6913, 6918, 6920, 6921, 6923, 
      6928, 6930, 6937, 6938, 6939, 6940, 6941, 6942, 6949, 6950, 6951, 6952, 
      6953, 6954, 6956, 6957, 6958, 6959, 6962, 6963, 6964, 6965, 6967, 6968, 
      6969, 6970, 6971, 6973, 6978, 6980, 6982, 6983, 6984, 6985, 6988, 6989, 
      6990, 6991, 6993, 6995, 6996, 6998, 6999, 7000, 7001, 7002, 7003, 7005, 
      7010, 7012, 7013, 7015, 7016, 7018, 7019, 7020, 7021, 7022, 7023, 7025, 
      7030, 7032, 7033, 7034, 7036, 7037, 7038, 7040, 7047, 7049, 7050, 7051, 
      7053, 7054, 7061, 7062, 7063, 7065, 7066, 7067, 7068, 7069, 7071, 7072, 
      7073, 7075, 7080, 7082, 7083, 7084, 7086, 7087, 7088, 7089, 7090, 7092, 
      7093, 7094, 7095, 7096, 7097, 7098, 7099, 7101, 7106, 7108, 7109, 7111, 
      7112, 7114, 7115, 7116, 7117, 7118, 7119, 7121, 7126, 7128, 7129, 7131, 
      7132, 7134, 7136, 7137, 7138, 7139, 7142, 7143, 7144, 7145, 7147, 7148, 
      7149, 7150, 7151, 7153, 7158, 7160, 7162, 7163, 7164, 7165, 7168, 7169, 
      7170, 7171, 7173, 7174, 7175, 7176, 7177, 7179, 7180, 7182, 7183, 7185, 
      7186, 7188, 7189, 7190, 7191, 7192, 7193, 7194, 7195, 7196, 7197, 7199, 
      7200, 7202, 7203, 7205, 7206, 7208, 7209, 7210, 7211, 7212, 7213, 7214, 
      7215, 7216, 7231, 7233, 7238, 7240, 7241, 7242, 7243, 7244, 7250, 7251, 
      7252, 7253, 7255, 7256, 7257, 7258, 7259, 7260, 7262, 7263, 7264, 7265, 
      7269, 7271, 7272, 7274, 7275, 7276, 7277, 7278, 7280, 7281, 7282, 7283, 
      7285, 7286, 7287, 7288, 7289, 7290, 7291, 7292, 7294, 7295, 7296, 7297, 
      7299, 7300, 7301, 7302, 7303, 7305, 7306, 7308, 7309, 7310, 7311, 7312, 
      7327, 7329, 7334, 7336, 7337, 7338, 7339, 7340, 7346, 7347, 7348, 7349, 
      7351, 7352, 7353, 7354, 7355, 7356, 7358, 7359, 7360, 7361, 7368, 7369, 
      7370, 7371, 7373, 7374, 7375, 7376, 7377, 7378, 7380, 7381, 7382, 7383, 
      7390, 7391, 7392, 7393, 7395, 7396, 7397, 7398, 7399, 7400, 7402, 7403, 
      7404, 7405, 7410, 7411, 7412, 7413, 7416, 7417, 7418, 7419, 7421, 7422, 
      7423, 7424, 7425, 7426, 7427, 7428, 7429, 7430, 7431, 7432, 7433, 7434, 
      7435, 7436, 7438, 7439, 7440, 7441, 7444, 7445, 7446, 7447, 7449, 7450, 
      7451, 7452, 7453, 7454, 7455, 7456, 7457, 7458, 7459, 7460, 7461, 7462, 
      7463, 7464, 7465, 7466, 7467, 7468, 7469, 7470, 7471, 7472, 7473, 7474, 
      7475, 7476, 7477, 7478, 7479, 7480, 7481, 7482, 7483, 7484, 7485, 7486, 
      7487, 7488, 7489, 7490, 7491, 7492, 7493, 7494, 7495, 7496, 7498, 7499, 
      7500, 7501, 7504, 7505, 7506, 7507, 7509, 7510, 7511, 7512, 7513, 7514, 
      7515, 7516, 7517, 7518, 7519, 7520, 7521, 7522, 7523, 7524, 7526, 7527, 
      7528, 7529, 7532, 7533, 7534, 7535, 7540, 7541, 7542, 7543, 7545, 7546, 
      7547, 7548, 7549, 7550, 7552, 7553, 7554, 7555, 7562, 7563, 7564, 7565, 
      7567, 7568, 7569, 7570, 7571, 7572, 7574, 7575, 7576, 7577, 7583, 7584, 
      7585, 7586, 7587, 7589, 7594, 7596, 7611, 7612, 7613, 7614, 7639, 7640, 
      7641, 7642, 7657, 7659, 7664, 7666, 7667, 7668, 7669, 7670}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[funny]\)], "Input"],

Cell[BoxData[
    \(3360\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ result5]]\)], "Input"],

Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ all5]]\)], "Input"],

Cell[BoxData[
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        12}}\)], "Output"]
}, Closed]],

Cell["\<\
the funny powers are found as power 2 and 3  of a selection of 1440  from the \
2200 matrices having powerlength 4,
and as powers 2,3,6,7,10 and 11 of a selection of 1920 from the 2880 matrices \
of powerlength 12.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ \((result5[\([funny]\)])\)]]\)], "Input"],

Cell[BoxData[
    \({{1440, 4}, {1920, 12}}\)], "Output"]
}, Closed]],

Cell["these are fully pseudoantisymmetric over all their powers:", "Text"],

Cell[BoxData[
    \(\(fullypseudoAS = Complement[Range[7672], funny];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[
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Cell[BoxData[
    \({{1, 1}, {31, 2}, {280, 3}, {760, 4}, {2280, 6}, {960, 12}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
            result5]]\)], "Input"],

Cell[BoxData[
    \({{312, 5}, {1600, 6}, {1920, 7}, {1440, 9}, {1920, 10}, {480, 
        13}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ all5]]\)], "Input"],

Cell[BoxData[
    \({{392, 5}, {1840, 6}, {1920, 7}, {2880, 9}, {3840, 10}, {480, 
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ Flatten[#])\) &\)\  /@ \ result5]]\)], "Input"],

Cell[BoxData[
    \({{1, \(-5\)}, {40, \(-4\)}, {385, \(-3\)}, {880, \(-2\)}, {1690, \
\(-1\)}, {1680, 0}, {1690, 1}, {880, 2}, {385, 3}, {40, 4}, {1, 
        5}}\)], "Output"]
}, Closed]],

Cell["\<\
Aha, there also are 1140 and 1920  matrices with 9 or 10 non-zero elements:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ 
                  Abs[Flatten[#]])\) &\)\  /@ \ \
\((result5[\([funny]\)])\)]]\)], "Input"],

Cell[BoxData[
    \({{1440, 9}, {1920, 10}}\)], "Output"]
}, Closed]],

Cell[BoxData[
    \(\(sum9 = 
        Select[result5[\([funny]\)], \((Plus @@ Abs[Flatten[#]])\) === 
              9 &];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[sum9]\)], "Input"],

Cell[BoxData[
    \(1440\)], "Output"]
}, Closed]],

Cell["\<\
and, Aha again, all funny ones have 9 or 10  non-zero elements, and vice \
versa : all those with 9 or 10 non-zero elements are funny:
those with 9 have powerlength 4, and their powers 2 & 3 are funny;
those with 10 have powerlength 12, and their powers 2,3,6,7,10 and 11 are \
funny (not PseudoAntiSymmetric);\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ sum9]]\)], "Input"],

Cell[BoxData[
    \({{1440, 4}}\)], "Output"]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[
          powerlength\  /@ 
            Complement[result5[\([funny]\)], \ sum9]]]\)], "Input"],

Cell[BoxData[
    \({{1920, 12}}\)], "Output"]
}, Closed]],

Cell["\<\
sorted form :  two sets of  130 matrices with equivalent sorted forms, 4 sets \
of 120 equivalents,  1 set of 105 equivalents ...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ \((\ \ Split[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ \((\ \ Split[
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              Split[Sort[\ \ \ sortedform\ \  /@ \ \((sum9)\)\ \ ]]\ \ ]]\ \
)\)\)], "Input"],

Cell[BoxData[
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ \((\ \ Split[
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              Split[Sort[\ \ \ sortedform\ \  /@ \ \((Complement[
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                        sum9])\)\ \ ]]\ \ ]]\ )\)\)], "Input"],

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}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["n=6", "Subsubsection"],

Cell["see notebook make_them.nb", "Text"],

Cell[BoxData[
    \(\(result6 = \ << \
"\<c:/Old_Disk_21_01_2003/_Wouter/result6.txt\>";\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[result6]\)], "Input"],

Cell[BoxData[
    \(83944\)], "Output"]
}, Open  ]],

Cell["do all powers themselves belong?", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result6, \((q_?MatrixQ)\) /; 
        Union[belongs /@ powerlist[q]] =!= {True}]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(0\)], "Output"]
}, Open  ]],

Cell["\<\
do the powers themselves produce extra (-1,0,1)-matrices whose powers are \
same, but non-pseudoantisymmetric ones?\
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
      result6, \((q_?MatrixQ)\) /; 
        Union[pseudoantisymmetric /@ powerlist[q]] =!= {True}]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(38400\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(powers6 = 
        Select[Union[Join @@ \((powerlist /@ \((result6)\))\)], 
          pseudoantisymmetric[#] === False &];\)\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[powers6]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(40260\)], "Output"]
}, Open  ]],

Cell[" so these are all (-1,0,1)-matrices whose powers are same:", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\(all6 = Join[result6, powers6];\)\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[all6]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(124204\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[Union@all6]\)], "Input",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(124204\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MemberQ[result6, IdentityMatrix[6]]\)], "Input"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[Det\  /@ \ result6]]\)], "Input"],

Cell[BoxData[
    \({{30872, \(-1\)}, {53072, 1}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ Split[Sort[Det\  /@ all6]]\)], "Input"],

Cell[BoxData[
    \({{50072, \(-1\)}, {74132, 1}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\ \(CoefficientList[CharacteristicPolynomial[#, x], 
                x] &\)\  /@ \ result6\ ]]\)], "Input"],

Cell[BoxData[
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          1}}, {240, {\(-1\), \(-3\), \(-3\), 0, 3, 3, 
          1}}, {1440, {\(-1\), \(-2\), \(-2\), 0, 2, 2, 
          1}}, {1080, {\(-1\), \(-2\), \(-1\), 0, 1, 2, 
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          1}}, {240, {\(-1\), \(-1\), 1, 0, \(-1\), 1, 1}}, {7560, {\(-1\), 
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          1}}, {360, {1, \(-1\), \(-1\), 2, \(-1\), \(-1\), 
          1}}, {3600, {1, \(-1\), 1, \(-2\), 1, \(-1\), 
          1}}, {2880, {1, \(-1\), 2, \(-1\), 2, \(-1\), 
          1}}, {3600, {1, \(-1\), 3, \(-2\), 3, \(-1\), 1}}, {1620, {1, 
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          1, \(-1\), \(-2\), \(-1\), 1, 1}}, {3600, {1, 1, 1, 2, 1, 1, 
          1}}, {2880, {1, 1, 2, 1, 2, 1, 1}}, {3600, {1, 1, 3, 2, 3, 1, 
          1}}, {15, {1, 2, \(-1\), \(-4\), \(-1\), 2, 1}}, {1440, {1, 2, 2, 
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          2, 1}}, {60, {1, 3, 3, 2, 3, 3, 1}}, {3600, {1, 3, 5, 6, 5, 3, 
          1}}, {960, {1, 3, 6, 7, 6, 3, 1}}, {270, {1, 4, 7, 8, 7, 4, 
          1}}, {720, {1, 4, 8, 10, 8, 4, 1}}, {60, {1, 5, 11, 14, 11, 5, 
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}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
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      Split[Sort[\ \(CoefficientList[CharacteristicPolynomial[#, x], 
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Cell[TextData[{
  "Would it be sufficient to show that for some A and B,  \
CharacteristicEqn(A)=CharacteristicEqn(B) \nbut Tr(A) <> Tr(B) to illustrate \
that De Rham's conjecture is false? ",
  ButtonBox["De Rham",
    ButtonData:>{
      URL[ "http://nyjm.albany.edu:8000/~mark/derham.html"], None},
    ButtonStyle->"Hyperlink"],
  "  (<A HREF=http://nyjm.albany.edu:8000/~mark/derham.html> De Rham </A> )"
}], "Text"],

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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["All resulting matrices are pseudo-antisymmetric,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["MatrixForm[Plus@@result6]", "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["MatrixForm[Plus@@Abs[result6]]", "Input"],

Cell[BoxData[
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            {"16536", "16536", "16536", "16536", "16536", "60064"}
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Cell["\<\
but their powers need not all be pseudo-antisymmetric !  38400 matrices of \
the total 83944  have at least one 'funny' power:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Count[
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Cell[BoxData[
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Cell["the indices of the funny ones are:", "Text"],

Cell[BoxData[
    \(\(funny = Union[First /@ %14];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Length[funny]\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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      Split[Sort[powerlength\  /@ \ compact6]]\)], "Input"],

Cell[BoxData[
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Cell["\<\
the funny powers are found as power 2 and 3  of a selection of 1440  from the \
2200 matrices having powerlength 4,
and as powers 2,3,6,7,10 and 11 of a selection of 1920 from the 2880 matrices \
of powerlength 12.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ \((result6[\([funny]\)])\)]]\)], "Input"],

Cell[BoxData[
    \({{15360, 4}, {23040, 12}}\)], "Output"]
}, Open  ]],

Cell["these are fully pseudoantisymmetric over all their powers:", "Text"],

Cell[BoxData[
    \(\(fullypseudoAS = Complement[Range[83944], funny];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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      Split[Sort[\(\((Plus @@ Abs[Flatten[#]])\) &\)\  /@ \ 
            result6]]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell["\<\
Aha, there also are 1140 and 1920  matrices with 9 or 10 non-zero elements:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[\(\((Plus @@ 
                  Abs[Flatten[#]])\) &\)\  /@ \ \
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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
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              10 &];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
and, Aha again, all funny ones have 9 or 10  non-zero elements, and vice \
versa : all those with 9 or 10 non-zero elements are funny:
those with 9 have powerlength 4, and their powers 2 & 3 are funny;
those with 10 have powerlength 12, and their powers 2,3,6,7,10 and 11 are \
funny (not PseudoAntiSymmetric);\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\({Length[#], First[#]} &\) /@ 
      Split[Sort[powerlength\  /@ \ sum10]]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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            Complement[result6[\([funny]\)], \ sum10]]]\)], "Input"],

Cell[BoxData[
    \({{5760, 4}, {23040, 12}}\)], "Output"]
}, Open  ]]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Irreducibility", "Subsection"],

Cell["irred1=result1;", "Input"],

Cell[CellGroupData[{

Cell["\<\
irred2=Cases[result2,q_?MatrixQ/;MatchQ[Abs[q],{{_,1},{1,_}}]];
Length[%]\
\>", "Input"],

Cell[BoxData[
    \(10\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
irred3=Cases[result3,q_?MatrixQ/;!MatchQ[Abs[q],{{_,_,_},{_,_,_},{0,0,_}}|{{_,\
_,_},{0,_,_},{0,_,_}}]];
Length[%]\
\>", "Input"],

Cell[BoxData[
    \(44\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
irred4=Cases[result4,q_?MatrixQ/;!MatchQ[Abs[q],
triamat[{_,   0,_,   0,_,_,   0,_,_,_}]|
triamat[{_,   _,_,   _,_,_,   0,0,0,_}]|
triamat[{_,   _,_,   0,0,_,   0,0,_,_}]
 ]];
Length[%]\
\>", "Input"],

Cell[BoxData[
    \(384\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
irred5=Cases[result5,q_?MatrixQ/;!MatchQ[Abs[q],
triamat[{_,   0,_,   0,_,_,   0,_,_,_,   0,_,_,_,_}]|
triamat[{_,   _,_,   _,_,_,   _,_,_,_,   0,0,0,0,_}]|
triamat[{_,   _,_,   0,0,_,   0,0,_,_,   0,0,_,_,_}]|
triamat[{_,   _,_,   _,_,_,   0,0,0,_,   0,0,0,_,_}]
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Length[%]\
\>", "Input"],

Cell[BoxData[
    \(3776\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\({irred1, irred2, irred3, irred4, 
          irred5}\  >> \ "\<H:/Data/Notebooks/irreducibles.txt\>";\)\)], \
"Input"],

Cell["only the irreducible matrices:", "Text"],

Cell[CellGroupData[{

Cell["{2,10,44,384,3776,43664}/2", "Input"],

Cell[BoxData[
    \({1, 5, 22, 192, 1888, 21832}\)], "Output"]
}, Open  ]],

Cell["the predictible block compositions from lower orders:", "Text"],

Cell[CellGroupData[{

Cell[" {2,14,92,796,7672,83944}-{2,10,44,384,3776,43664}", "Input"],

Cell[BoxData[
    \({0, 4, 48, 412, 3896, 40280}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
{0,2*2,2*2*2+2*10*(2),2*2*2*2+2*2*10*(3)+2*44*(2)+10*10,
2*2*2*2*2+2*2*2*10*(4)+2*2*44*(3)+2*10*10*(3)+10*44*(2)+2*384*(2),
2^6+2^4*10*(5)+2*2*2*44*(4)+2*10*44*(6)+2*2*10*10*(4!/2!/2!)+10*10*10+44*44+\
10*384*(2)+2*2*384*(3)+2*3776*(2)}
\
\>", "Input"],

Cell[BoxData[
    \({0, 4, 48, 412, 3896, 40280}\)], "Output"]
}, Open  ]],

Cell["result6= <<\"c:/Old_Disk_21_01_2003/_Wouter/result6.txt\";", "Input"],

Cell[CellGroupData[{

Cell["Length[result6]", "Input"],

Cell[BoxData[
    \(83944\)], "Output"]
}, Open  ]],

Cell["\<\
irred6=Cases[result6,q_?MatrixQ/;!MatchQ[Abs[q],
triamat[{_,   0,_,   0,_,_,   0,_,_,_,   0,_,_,_,_,   0,_,_,_,_,_}]|
triamat[{_,   _,_,   _,_,_,   _,_,_,_,   _,_,_,_,_,   0,0,0,0,0,_}]|
triamat[{_,   _,_,   0,0,_,   0,0,_,_,   0,0,_,_,_,   0,0,_,_,_,_}]|
triamat[{_,   _,_,   _,_,_,   _,_,_,_,   0,0,0,0,_,   0,0,0,0,_,_}]|
triamat[{_,   _,_,   _,_,_,   0,0,0,_,   0,0,0,_,_,   0,0,0,_,_,_}]
 ]];\
\>", "Input"],

Cell[CellGroupData[{

Cell["Length[irred6]", "Input"],

Cell[BoxData[
    \(43664\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\((compact6 = \((Union[
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                      Sort[{#, Transpose[#], \(-Transpose[#]\), 
                          Reverse[
                            Reverse /@ #], \(-#\), \(-Reverse[
                              Reverse /@ #]\)}]] &\) /@ \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["Cayley-Hamilton Theorem", "Subsection"],

Cell[CellGroupData[{

Cell["for n=4 :", "Subsubsection",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)