(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 9.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1063, 20] NotebookDataLength[ 70154, 1475] NotebookOptionsPosition[ 68841, 1406] NotebookOutlinePosition[ 69347, 1428] CellTagsIndexPosition[ 69304, 1425] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["A235647 sequence proposal", "Section"], Cell[TextData[StyleBox["Number of chains in the sorting permutation of the \ (a,b) 2D finite sequence \ {(1,1),(1,2),...,(1,n),...,(k,k),(k,k+1),...,(k,n),...,(n-1,n-1),(n-1,n),(n,n)\ }\nto b-ascending and then a-ascending", FontSize->18]], "Text", Background->RGBColor[0.88, 1, 0.88]], Cell[CellGroupData[{ Cell["Description", "Subsection"], Cell["\<\ For a specific positive integer n, we define two 2D finite sequences A and B \ as follows: A={(a,b) where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is sorted \ a-ascending and then b-ascending} B=(A sorted b-ascending and then a-ascending) One can easily deduce that: B={(a,b) where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is sorted \ b-ascending and then a-ascending} Since A and B sequences contain exactly the same members there is a 1-1 \ correspondence between them that describes the effect of the sorting A to get \ B. This correspondence is in fact a permutation of their positions in A and B. A235647(n) equals the number of chains (including 1-cycles) in the \ permutation that corresponds A elements to B elements\ \>", "Text"], Cell["\<\ Example: A and B sequences for n=6\ \>", "Text", Background->RGBColor[1, 1, 0.85]], Cell[BoxData[ RowBox[{ RowBox[{"n", "=", "6"}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"A", "=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "a", ",", "n"}], "}"}]}], "]"}], ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"B", "=", RowBox[{"Flatten", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "1", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "b"}], "}"}]}], "]"}], ",", "1"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Grid", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\"\<A\>\"", ",", "\"\<B\>\""}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"TableForm", "@", "A"}], ",", RowBox[{"TableForm", "@", "B"}]}], "}"}]}], "}"}], ",", RowBox[{"Frame", "\[Rule]", "All"}]}], "]"}]}], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"A\"\>", "\<\"B\"\>"}, { TagBox[GridBox[{ {"1", "1"}, {"1", "2"}, {"1", "3"}, {"1", "4"}, {"1", "5"}, {"1", "6"}, {"2", "2"}, {"2", "3"}, {"2", "4"}, {"2", "5"}, {"2", "6"}, {"3", "3"}, {"3", "4"}, {"3", "5"}, {"3", "6"}, {"4", "4"}, {"4", "5"}, {"4", "6"}, {"5", "5"}, {"5", "6"}, {"6", "6"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[2.0999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]], TagBox[GridBox[{ {"1", "1"}, {"1", "2"}, {"2", "2"}, {"1", "3"}, {"2", "3"}, {"3", "3"}, {"1", "4"}, {"2", "4"}, {"3", "4"}, {"4", "4"}, {"1", "5"}, {"2", "5"}, {"3", "5"}, {"4", "5"}, {"5", "5"}, {"1", "6"}, {"2", "6"}, {"3", "6"}, {"4", "6"}, {"5", "6"}, {"6", "6"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[2.0999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]]} }, AutoDelete->False, GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Permutation calculation", "Subsubsection"], Cell["\<\ Lemma: (a,b) element in B sequence is at position (b-1)b/2+a. Proof B is sorted b-ascending and then a-ascending, thus (a,b) is a places after \ (b-1,b-1)\[CloseCurlyQuote]s position. But (b-1,b-1) is at position 1+2+...+(b-1)= (b-1)b/2 QED \ \>", "Text"], Cell["\<\ Concequently the permutation for n is P={(b-1)b/2+a where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is \ sorted a-ascending and then b-ascending (as in A)} \ \>", "Text"], Cell["\<\ Example: Permutation calculation for n=6\ \>", "Text", Background->RGBColor[1, 1, 0.85]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"P", "=", RowBox[{"Flatten", "@", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"b", "-", "1"}], ")"}], RowBox[{"b", "/", "2"}]}], "+", "a"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "n"}], "}"}], ",", RowBox[{"{", RowBox[{"b", ",", "a", ",", "n"}], "}"}]}], "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "2", ",", "4", ",", "7", ",", "11", ",", "16", ",", "3", ",", "5", ",", "8", ",", "12", ",", "17", ",", "6", ",", "9", ",", "13", ",", "18", ",", "10", ",", "14", ",", "19", ",", "15", ",", "20", ",", "21"}], "}"}]], "Output"] }, Open 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