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"\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`permLength$$], 10, "permutation length"}, 2, 40, 1}, {{ Hold[$CellContext`sr$$], 52508, "new random permutation of the same length"}, 1, 123456, 1}}, Typeset`size$$ = {450., {246., 115.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`permLength$47046$$ = 0, $CellContext`sr$47047$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`permLength$$ = 10, $CellContext`sr$$ = 52508}, "ControllerVariables" :> { Hold[$CellContext`permLength$$, $CellContext`permLength$47046$$, 0], Hold[$CellContext`sr$$, $CellContext`sr$47047$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Module[{$CellContext`perm = (SeedRandom[$CellContext`sr$$]; RandomSample[ Range[$CellContext`permLength$$], $CellContext`permLength$$]), \ $CellContext`c}, $CellContext`c = $CellContext`cycles[$CellContext`perm]; Labeled[ GraphPlot[ $CellContext`cyclesToGraph[$CellContext`c], DirectedEdges -> True, VertexLabeling -> True, ImageSize -> {450, 225}, AspectRatio -> 225/450], Pane[ Text[ Grid[{{"permutation", $CellContext`perm}, {"cycles", $CellContext`cycleNotation[$CellContext`c]}}, Background -> {{ ColorData["Legacy"]["DeepNaplesYellow"], Lighter[ ColorData["Legacy"]["DeepNaplesYellow"], 0.3]}, None}, ItemSize -> {{8, 26}, 1}, Dividers -> All, BaseStyle -> {14}]], { 450, 130}]]], "Specifications" :> {{{$CellContext`permLength$$, 10, "permutation length"}, 2, 40, 1, Appearance -> "Labeled"}, {{$CellContext`sr$$, 52508, "new random permutation of the same length"}, 1, 123456, 1}}, "Options" :> {}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{495., {236., 241.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(({$CellContext`cycles[ Pattern[$CellContext`perm, Blank[]]] := First[ Last[ Reap[ NestWhile[Complement[#, Sow[ Most[ NestWhileList[Part[$CellContext`perm, #]& , First[#], UnsameQ, All]]]]& , $CellContext`perm, # =!= {}& ]]]], \ $CellContext`cyclesToGraph[ Pattern[$CellContext`cycleStructure, Blank[List]]] := Flatten[ Apply[Rule, Map[Partition[#, 2, 1, {1, 1}]& , $CellContext`cycleStructure], { 2}]], $CellContext`cycleNotation[ Pattern[$CellContext`cycle, Blank[]]] := StringTake[ StringReplace[ ToString[$CellContext`cycle], { "{" -> "(", RegularExpression["},?"] -> ")"}], {2, -2}]}; Typeset`initDone$$ = True); ReleaseHold[ HoldComplete[{$CellContext`cycles[ Pattern[$CellContext`perm, Blank[]]] := First[ Last[ Reap[ NestWhile[Complement[#, Sow[ Most[ NestWhileList[Part[$CellContext`perm, #]& , First[#], UnsameQ, All]]]]& , $CellContext`perm, # =!= {}& ]]]], \ $CellContext`cyclesToGraph[ Pattern[$CellContext`cycleStructure, Blank[List]]] := Flatten[ Apply[Rule, Map[Partition[#, 2, 1, {1, 1}]& , $CellContext`cycleStructure], { 2}]], $CellContext`cycleNotation[ Pattern[$CellContext`cycle, Blank[]]] := StringTake[ StringReplace[ ToString[$CellContext`cycle], { "{" -> "(", RegularExpression["},?"] -> ")"}], {2, -2}]}]]; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->180252204], Cell[TextData[{ "A permutation is a list in which each element occurs only once. If the \ members of the permutation have values ", Cell[BoxData[ FormBox[ RowBox[{"1", ",", "\[Ellipsis]", ",", "n"}], TraditionalForm]], "InlineMath"], ", where ", Cell[BoxData[ FormBox["n", TraditionalForm]], "InlineMath"], " is the length of the permutation, then you can think of a permutation as \ an \"onto\" (\"surjective\") function of the set ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"1", ",", "2", ",", "\[Ellipsis]", ",", "n"}], "}"}], TraditionalForm]], "InlineMath"], " to itself. If you successively apply this function to any member of the \ set, you obtain its \"cycle\" or \"orbit\". This Demonstration shows how you \ can write a permutation as a list of these cycles and how these cycles can be \ visualized as a graph. 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