Tableau Promotion

Author

Eric W. Weisstein
August 29, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Combinatorics/TableauPromotion.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/TableauPromotion.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Illustration

Promotion of a standard Young tableau of shape (3, 2).

Module[{arrowColor, labelBackground, neutral, offsets, stages, tableauPrimitives},
	neutral = LightDarkSwitched[Black, White]; labelBackground = LightDarkSwitched[White, Black]; arrowColor = RGBColor[0.18, 0.48, 0.75];
	stages = {{{1, 2, 5}, {3, 4}}, {{None, 2, 5}, {3, 4}}, {{2, 4, 5}, {3, None}}, {{1, 3, 4}, {2, 5}}}; offsets = {0, 4.7, 9.4, 14.1};
	tableauPrimitives[tab_, x_] := Flatten[MapIndexed[Function[{row, ri}, MapIndexed[Function[{entry, ci}, With[{p = {x + ci[[1]] - 1, 1 - ri[[1]]}}, {EdgeForm[Directive[neutral, AbsoluteThickness[1.4]]], FaceForm[None], Rectangle[p, p + {1, 1}], If[entry === None, Nothing, Text[Style[entry, 18, neutral, FontFamily -> "Times"], p + {0.5, 0.5}]]}]], row]], tab]];
	Graphics[{tableauPrimitives @@@ Transpose[{stages, offsets}], Directive[arrowColor, AbsoluteThickness[2]], Arrow[{{3.25, 0}, {4.35, 0}}], Arrow[{{7.95, 0}, {9.05, 0}}], Arrow[{{12.65, 0}, {13.75, 0}}], Text[Style["erase 1", 13, neutral, Background -> labelBackground, FontFamily -> "Times"], {3.8, 1.55}], Text[Style["slide 2, then 4", 13, neutral, Background -> labelBackground, FontFamily -> "Times"], {8.5, 1.55}], Text[Style["decrease; insert 5", 13, neutral, Background -> labelBackground, FontFamily -> "Times"], {13.2, 1.55}], Text[Style["T", 17, neutral, FontFamily -> "Times", FontSlant -> "Italic"], {1.5, -1.6}], Text[Style[Row[{"promotion", "(", Style["T", FontSlant -> "Italic"], ")"}], 17, neutral, FontFamily -> "Times"], {15.6, -1.6}]}, PlotRange -> {Automatic, {-2.4, 2.4}}]
]