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"\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "6"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "7"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "3"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "5"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "7"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "9"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}]}]], "Print"], Cell[BoxData[ RowBox[{ RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "3"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "4"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"6", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "5"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "6"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{"3", " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "7"], " ", SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]\[Prime]", MultilineFunction->None], "[", "p", "]"}], "2"]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "3"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "5"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "7"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}], "+", RowBox[{ SuperscriptBox[ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "9"], " ", RowBox[{ SuperscriptBox["f", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "p", "]"}]}]}]], "Print"], Cell[BoxData[ \(0\)], "Print"] }, Open ]], Cell["\<\ The following routine validates the Schroeder summations by computing the \ error between the Schroeder summations and the derivatives of iterated \ functions where k is the range of derivatives tested and n is the number of \ iterations. This proves that the Schroeder summations always properly give \ the first eight derivatives of the first eight iterates of f.\ \>", "Text", Background->None], Cell[BoxData[GridBox[{ {\(\(D\^1\) p\), "\[Ellipsis]", \(\(D\^1\) \(f\^8\)[p]\)}, {\(\(D\^2\) p\), " ", \(\(D\^2\) \(f\^8\)[p]\)}, {"\[Ellipsis]", " ", "\[Ellipsis]"}, {\(\(D\^7\) p\), " ", \(\(D\^7\) \(f\^8\)[p]\)}, {\(\(D\^8\) p\), "\[Ellipsis]", \(\(D\^8\) \(f\^8\)[p]\)} }]], "Text", Background->None], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"t", "=", RowBox[{"Table", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\(Dyne[i]\), "/.", RowBox[{ RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}], "\[Rule]", " ", "a"}]}], " ", "/.", " ", RowBox[{"a", " ", "\[Rule]", " ", RowBox[{ SuperscriptBox["f", "\[Prime]", MultilineFunction->None], "[", "p", "]"}]}]}], ")"}], "-", \((D[\ Nest[f, p, n], {p, i}]\ //. \ f[p] \[Rule] p)\)}], "\[IndentingNewLine]", ",", \({i, 1, size}\), "\[IndentingNewLine]", ",", \({n, 0, 8}\)}], "\[IndentingNewLine]", "]"}]}], ";"}], "\[IndentingNewLine]", \(\(t // ExpandAll\)\ // TableForm\)}], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Labled Hierarchies", "Subsection", Background->None], Cell["\<\ This adds up the coefficients of the nth Schroeder summation giving the \ hierarchies of n just as adding up the coefficients of the nth Bell summation \ gives the nth Bell number.\ \>", "Text", Background->None], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[\(dyne[i] /. dyn[_] \[Rule] 1\) /. d[_] \[Rule] 1], {i, size}]\)], "Input", Background->None], Cell[BoxData[ \(1\)], "Print"], Cell[BoxData[ \(1\)], "Print"], Cell[BoxData[ \(4\)], "Print"], Cell[BoxData[ \(26\)], "Print"], Cell[BoxData[ \(236\)], "Print"], Cell[BoxData[ \(2752\)], "Print"], Cell[BoxData[ \(39208\)], "Print"], Cell[BoxData[ \(660032\)], "Print"] }, Open ]], Cell["\<\ The following is an uses the pointing operator to generate the instances of \ labeled hierarchies. Only the first six values of the hierarchies structure \ are computed due to time. Even with the algorithmic complexity of computing \ Schroder summations, they can be used to calculate the values of the \ hierarchies structure faster than using the pointing operator because the \ Schroder summations are actually generating the unlabeled heirachies \ combinatoric structure which grows much for sloly than the labeled version.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \( (*\ Enumerate\ the\ instances\ of\ the\ hierachies\ of\ four\ items\ *) \ \[IndentingNewLine]hierarchies[4]\[IndentingNewLine]\[IndentingNewLine] (*\ Display\ the\ the\ first\ six\ values\ of\ the\ hierachies\ \ \(\(structure\)\(.\)\)\ *) \), "\[IndentingNewLine]", \(Do[Print[hierarchies[i] /. hier[_] \[Rule] 1], {i, 6}]\)}], "Input"], Cell[BoxData[ \(hier[{1, {2, {3, 4}}}] + hier[{1, {{2, 3}, 4}}] + hier[{1, {{2, 4}, 3}}] + hier[{1, {2, 3, 4}}] + hier[{{1, 2}, {3, 4}}] + hier[{{1, 3}, {2, 4}}] + hier[{{1, 4}, {2, 3}}] + hier[{{1, {2, 3}}, 4}] + hier[{{1, {2, 4}}, 3}] + hier[{{1, {3, 4}}, 2}] + hier[{{{1, 2}, 3}, 4}] + hier[{{{1, 2}, 4}, 3}] + hier[{{{1, 3}, 2}, 4}] + hier[{{{1, 3}, 4}, 2}] + hier[{{{1, 4}, 2}, 3}] + hier[{{{1, 4}, 3}, 2}] + hier[{{1, 2, 3}, 4}] + hier[{{1, 2, 4}, 3}] + hier[{{1, 3, 4}, 2}] + hier[{1, 2, {3, 4}}] + hier[{1, {2, 3}, 4}] + hier[{1, {2, 4}, 3}] + hier[{{1, 2}, 3, 4}] + hier[{{1, 3}, 2, 4}] + hier[{{1, 4}, 2, 3}] + hier[{1, 2, 3, 4}]\)], "Output"], Cell[BoxData[ \(1\)], "Print"], Cell[BoxData[ \(1\)], "Print"], Cell[BoxData[ \(4\)], "Print"], Cell[BoxData[ \(26\)], "Print"], Cell[BoxData[ \(236\)], "Print"], Cell[BoxData[ \(2752\)], "Print"] }, Open ]], Cell["Unlabled Hierarchies", "Subsection", Background->None], Cell["\<\ This adds up the number of terms of the nth Schroeder summation giving the \ unlabeled hierarchies of n just as adding up the number of terms of the nth \ Bell summation gives the nth partition number. Both unlabeled structures \ serve as indexes to their label versions. The initial value is wrong due to \ the way the program is implemented, it should be one, not zero.\ \>", "Text", Background->None], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[Count[dyne[i], dyn[__], {1, \[Infinity]}]], {i, 1, size}]\)], "Input", Background->None], Cell[BoxData[ \(0\)], "Print"], Cell[BoxData[ \(1\)], "Print"], Cell[BoxData[ \(2\)], "Print"], Cell[BoxData[ \(5\)], "Print"], Cell[BoxData[ \(12\)], "Print"], Cell[BoxData[ \(33\)], "Print"], Cell[BoxData[ \(90\)], "Print"], Cell[BoxData[ \(261\)], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Hierarchies of Height n", "Subsection", Background->None], Cell[CellGroupData[{ Cell[BoxData[{ \(<< Iterate`\), "\[IndentingNewLine]", \(\(size = 8;\)\), "\[IndentingNewLine]", \(\(GfRoot[z_] := Exp[z] - 1;\)\), "\[IndentingNewLine]", \(\(Iterate[GfRoot, n, z, 0, size];\)\), "\[IndentingNewLine]", \(\(Print["\"];\ \)\), "\[IndentingNewLine]", \(Sum[ 1/\(k!\)\ Simplify[Dyne[k]]\ z\^k, {k, 1, size}]\), "\[IndentingNewLine]", \(\(Print["\"];\)\), "\[IndentingNewLine]", \(tree = Table[Simplify[Dyne[k]], {k, 1, size}]\), "\[IndentingNewLine]", \(\(hierHeight[i_, info_] := Print["\", i, info, "\<\n\>", \ tree\ /. \ n \[Rule] i, "\<\n\>"];\)\), "\[IndentingNewLine]", \(\(Print["\"];\)\), "\[IndentingNewLine]", \(hierHeight[0, "\<\>"]\), "\[IndentingNewLine]", \(hierHeight[1/4, "\<\>"]\), "\[IndentingNewLine]", \(hierHeight[1/3, "\<\>"]\), "\[IndentingNewLine]", \(hierHeight[1/2, "\< - 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3129\ n\^2 + 6125\ \ n\^3 - 5481\ n\^4 + 1890\ n\^5)\)\ z\^7\)\/120960 + \(n\ \((\(-22\) - 645\ n \ + 5474\ n\^2 - 15540\ n\^3 + 21014\ n\^4 - 14049\ n\^5 + 3780\ n\^6)\)\ \ z\^8\)\/483840\)], "Output"], Cell[BoxData[ \("Terms of the generating function of the hierarchies of height n."\)], \ "Print"], Cell[BoxData[ \({1, n, 1\/2\ n\ \((\(-1\) + 3\ n)\), 1\/2\ n\ \((1 - 5\ n + 6\ n\^2)\), 1\/6\ n\ \((\(-4\) + 30\ n - 65\ n\^2 + 45\ n\^3)\), 1\/24\ n\ \((22 - 273\ n + 890\ n\^2 - 1155\ n\^3 + 540\ n\^4)\), 1\/24\ n\ \((\(-18\) + 637\ n - 3129\ n\^2 + 6125\ n\^3 - 5481\ n\^4 + 1890\ n\^5)\), 1\/12\ n\ \((\(-22\) - 645\ n + 5474\ n\^2 - 15540\ n\^3 + 21014\ n\^4 - 14049\ n\^5 + 3780\ n\^6)\)}\)], "Output"], Cell[BoxData[ \("Axxxxxx are references to The On-Line Encyclopedia of Integer \ Sequences"\)], "Print"], Cell[BoxData[ InterpretationBox[\("Hierarchies Height: "\[InvisibleSpace]0\ \[InvisibleSpace]\*"\<\"\"\>"\[InvisibleSpace]"\n"\[InvisibleSpace]{1, 0, 0, 0, 0, 0, 0, 0}\[InvisibleSpace]"\n"\), SequenceForm[ "Hierarchies Height: ", 0, "", "\n", {1, 0, 0, 0, 0, 0, 0, 0}, "\n"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Hierarchies Height: "\[InvisibleSpace]1\/4\ \[InvisibleSpace]\*"\<\"\"\>"\[InvisibleSpace]"\n"\[InvisibleSpace]{1, 1\/4, \(-\(1\/32\)\), 1\/64, 3\/512, \(-\(35\/512\)\), 3725\/16384, \(-\(625\/2048\)\)}\[InvisibleSpace]"\n"\), SequenceForm[ "Hierarchies Height: ", Rational[ 1, 4], "", "\n", {1, Rational[ 1, 4], Rational[ -1, 32], Rational[ 1, 64], Rational[ 3, 512], Rational[ -35, 512], Rational[ 3725, 16384], Rational[ -625, 2048]}, "\n"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Hierarchies Height: "\[InvisibleSpace]1\/3\ \[InvisibleSpace]\*"\<\"\"\>"\[InvisibleSpace]"\n"\[InvisibleSpace]{1, 1\/3, 0, 0, 2\/81, \(-\(7\/81\)\), 46\/243, 50\/729}\[InvisibleSpace]"\n"\), SequenceForm[ "Hierarchies Height: ", Rational[ 1, 3], "", "\n", {1, Rational[ 1, 3], 0, 0, Rational[ 2, 81], Rational[ -7, 81], Rational[ 46, 243], Rational[ 50, 729]}, "\n"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Hierarchies Height: "\[InvisibleSpace]1\/2\ \[InvisibleSpace]" - 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FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}, FontSize->11], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Outline Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Outline1"], CellMargins->{{12, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 50}, ParagraphIndent->-38, CounterIncrements->"Outline1", FontSize->18, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}], Cell[StyleData["Outline1", "Printout"], CounterBoxOptions->{CounterFunction:>CapitalRomanNumeral}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline2"], CellMargins->{{59, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 60}, ParagraphIndent->-27, CounterIncrements->"Outline2", FontSize->15, FontWeight->"Bold", CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}], Cell[StyleData["Outline2", "Printout"], CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "A", "Z"], #]&)}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline3"], CellMargins->{{108, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 70}, ParagraphIndent->-21, CounterIncrements->"Outline3", FontSize->12, CounterBoxOptions->{CounterFunction:>Identity}], Cell[StyleData["Outline3", "Printout"], CounterBoxOptions->{CounterFunction:>Identity}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Outline4"], CellMargins->{{158, 10}, {7, 7}}, CellGroupingRules->{"SectionGrouping", 80}, ParagraphIndent->-18, CounterIncrements->"Outline4", FontSize->10, CounterBoxOptions->{CounterFunction:>(Part[ CharacterRange[ "a", "z"], #]&)}], Cell[StyleData["Outline4", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0