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See \ http://mathworld.wolfram.com/YoungTableau.html. (I won't explain here why \ they are useful-maybe you'll see them in another class. ) \[FilledRectangle] Associated with any decreasing sequence of natural \ numbers, such as (5,3,2,2,1), there is a Young frame, as below, whose row \ lengths are given by the sequence.\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 %%ImageSize: 100 100 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00952381 0.97619 0.00952381 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g 1 Mabswid [ ] 0 setdash .02381 .97619 m .97619 .97619 L s .02381 .78571 m .02381 .97619 L s .21429 .78571 m .21429 .97619 L s .40476 .78571 m .40476 .97619 L s .59524 .78571 m .59524 .97619 L s .78571 .78571 m .78571 .97619 L s .97619 .78571 m .97619 .97619 L s .02381 .78571 m .97619 .78571 L s .02381 .59524 m .02381 .78571 L s .21429 .59524 m .21429 .78571 L s .40476 .59524 m .40476 .78571 L s .59524 .59524 m .59524 .78571 L s .02381 .59524 m .59524 .59524 L s .02381 .40476 m .02381 .59524 L s .21429 .40476 m .21429 .59524 L s .40476 .40476 m .40476 .59524 L s .02381 .40476 m .40476 .40476 L s .02381 .21429 m .02381 .40476 L s .21429 .21429 m .21429 .40476 L s .40476 .21429 m .40476 .40476 L s .02381 .21429 m .40476 .21429 L s .02381 .02381 m .02381 .21429 L s .21429 .02381 m .21429 .21429 L s .02381 .02381 m .21429 .02381 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{100, 100}, ImageMargins->{{158, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["CompressedBitmap", "\<\ eJztmUEKwjAQRSfmALrwANmVHqG41416hRJaoV2IUnv/mIEuWsiiIrYgL5Am /Jm8twuBXnzf3O6+byvvTp1/Nm31csdHFyNrRKTW6UT3IW6HTx3iiIXJstFl rw3XUSBWkzAM2SWyM+200z6nPdPK4bPzqawAAQIEiHUQXGMgQIBYE1FoJf+e uU1kGViwYMGC/RdsqRX7G8/cswY9evTo0aNHj345fTl6/kx/+4l5A/4eH0o= \ \>"], ImageRangeCache->{{{165.438, 244.438}, {351.062, 272.062}} -> {-222.388, \ 259.105, 1.0633, 1.0633}}], Cell["\<\ \[FilledRectangle] A Young tableau is a Young frame in which the boxes are \ filled with the natural numbers 1,2,3,....in some way.\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! 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There are three components:\n\n\ 1) Cretion of the frame from the list of the values given\n2) Placing the \ numbers into the frame\n3) Checking whether the Young tableau is standard or \ semistandard or neither. \n\nFor the frame, the user shoul be able to change \ the size of the boxes, the thickness of the lines and the font. If no such \ items are given, then your function should use a pre-defined size, thickness \ and font." }], "Text", CellChangeTimes->{{3.420157528049437*^9, 3.420157723305285*^9}}], Cell[BoxData[ RowBox[{"yframe", "[", RowBox[{"{", RowBox[{"5", ",", "3", ",", "2", ",", "2", ",", "1"}], "}"}], "]"}]], "Input"], Cell["the Young tableau with a command such as ", "Text"], Cell[BoxData[ RowBox[{"ytableau", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "4", ",", "5", ",", "6", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "10", ",", "11"}], "}"}], ",", RowBox[{"{", RowBox[{"13", ",", "12"}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "8"}], "}"}], ",", RowBox[{"{", "9", "}"}]}], "}"}], "]"}]], "Input"], Cell["\<\ It would be useful to be able change the size of these diagrams, the \ thickness of the lines and the font. This could be done by defining certain \ options for your function: \ \>", "Text", CellChangeTimes->{ 3.420157778993058*^9, {3.4201578655937853`*^9, 3.420157871880039*^9}}], Cell["\<\ ystyle - the style for the font, for example {FontSize -> 18, FontFamily -> \ \"Impact\"} ysize - in pts, for example 24 ythick- in pts, for example 3; \ \>", "Text", CellChangeTimes->{{3.420157733369969*^9, 3.4201577373701067`*^9}, { 3.420157782230249*^9, 3.4201578012031927`*^9}, {3.420157875858349*^9, 3.420157938612033*^9}}], Cell[BoxData[ RowBox[{"ytableau", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ "1", ",", " ", "4", ",", " ", "5", ",", " ", "6", ",", " ", "2"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"3", ",", " ", "10", ",", " ", "11"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"13", ",", " ", "12"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"7", ",", " ", "8"}], "}"}], ",", " ", RowBox[{"{", "9", "}"}]}], "}"}], ",", " ", "ystyle", ",", " ", "ysize", ",", " ", "ythick"}], "]"}]], "Input", 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Of course, these diagrams will not all be 4in wide, so you \ will have to set ImageSize appropriately in your commands.\n\nYour functions \ should return the Young tableau together with a Print statement if the \ tableau is eitehr standard or semi-standard (this means that your function \ needs to check whether in each row/column the values are strictly increasing \ or non-decreasing. For this you want to use functions that operate on lists, \ such as ", StyleBox["Transpose ", FontWeight->"Bold"], "(to get the columns). \n\nFinally, you are to check that all your aguments \ are of the right form (i.e. there are no empty lists or negative values, and \ that there are all the values from 1 through n and no duplicates). Your \ function should produce an error statement of this is not the case, and there \ should be a usage function like the ones ", StyleBox["Mathematica", FontSlant->"Italic"], " gives when using the ? mechanism.\n\n\n", StyleBox["Note: ", FontWeight->"Bold"], " In the web site that was referenced above, the columns go upwards, but I \ want them downwards as in the example given above. 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