YoungToPoset
YoungToPoset
Convert a Young Tableau to a partially ordered set of coordinates
Definition
Definition
YoungToPoset[young_]:=Last/@Sort[Flatten[MapIndexed[{#1,#2}&,young,{2}],1]];
PosetToYoung[poset_]:=TakeList[Flatten[Position[poset,#]&/@Sort[poset]],Length/@SplitBy[Sort[poset],First]];
SkewPoset[poset_]:={#[[1]]+#[[2]]-1,#[[2]]}&/@poset;
UnskewPoset[skew_]:={#[[1]]-#[[2]]+1,#[[2]]}&/@skew;
Get["Combinatorica`"]
SkewPosetDiagram[skew_]:=With[{p={#2-#1/2,#1}&@@@-skew},Graphics[{Point[p],Orange,Arrowheads[Large],Arrow/@Partition[p,2,1]},ImageSizeTiny]];
Documentation
Documentation
Usage
Usage
YoungToPoset[young]
converts the Young tableau young to a partially ordered set of coordinates.
Details & Options
Details & Options
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Additional information about usage and options.
Examples
Examples
Basic Examples
Basic Examples
Some Young tableaux:
young=Combinatorica`Tableaux[{3,2,1}]
Convert to posets:
posets=YoungToPoset/@young
Convert to skew posets:
skew=SkewPoset/@posets
The Young tableaux along with skew poset diagrams:
Grid[Partition[Column[#,AlignmentCenter]&/@Transpose[{Grid[Reverse[#],FrameAll]&/@young,SkewPosetDiagram/@skew}],8],FrameAll]
A large random skew diagram:
SkewPosetDiagram[SkewPoset[YoungToPoset@Combinatorica`RandomTableau[{9,8,7,6,5,4,3,2,1}]]]
Scope
Scope
Partial ordering in the simple posets:
tab=Flatten[Table[{{i,j}{i+1,j},{i,j}{i,j+1}},{i,1,3},{j,1,3}]];vert=Union[Flatten[List@@@tab,1]];dg=DirectedGraph[tab,VertexLabelsAutomatic,VertexCoordinatesThread[vertvert]]
Partial ordering in the skew posets:
tab=Flatten[Table[{{i,j}{i+1,j},{i,j}{i+1,j+1}},{i,1,3},{j,1,i}]];vert=Union[Flatten[List@@@tab,1]];dg=DirectedGraph[tab,VertexLabelsAutomatic,VertexCoordinatesThread[vertvert]]
Options
Options
Applications
Applications
Properties and Relations
Properties and Relations
Possible Issues
Possible Issues
Neat Examples
Neat Examples
Source & Additional Information
Source & Additional Information
Ed Pegg Jr
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Young tableau
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Young tableaux
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Poset
Categories
Categories
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SymbolName (documented Wolfram Language symbol)
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Resource Name (resources from any Wolfram repository)
Source, reference or citation information
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Link to other related material
Author Notes
Author Notes
Additional information about limitations, issues, etc.
Additional information for the reviewer.