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allSemistandardTableaux -- list all semistandard Young tableaux of a given shape

Description

i1 : theBag = allSemistandardTableaux(new Partition from {5,4,2}, new Partition from {3,1})

o1 = -*a bagged sequence of length 144*-

o1 : Bag
 
i2 : for i from 0 to 4 do print theBag#i
            ┌───┬───┐
            │ 1 │ 1 │
    ┌───┬───┼───┼───┘
    │ 1 │ 1 │ 2 │    
┌───┼───┼───┴───┘    
│ 1 │ 2 │            
└───┴───┘            
            ┌───┬───┐
            │ 1 │ 1 │
    ┌───┬───┼───┼───┘
    │ 1 │ 1 │ 2 │    
┌───┼───┼───┴───┘    
│ 1 │ 3 │            
└───┴───┘            
            ┌───┬───┐
            │ 1 │ 1 │
    ┌───┬───┼───┼───┘
    │ 1 │ 1 │ 2 │    
┌───┼───┼───┴───┘    
│ 2 │ 2 │            
└───┴───┘            
            ┌───┬───┐
            │ 1 │ 1 │
    ┌───┬───┼───┼───┘
    │ 1 │ 1 │ 2 │    
┌───┼───┼───┴───┘    
│ 2 │ 3 │            
└───┴───┘            
            ┌───┬───┐
            │ 1 │ 1 │
    ┌───┬───┼───┼───┘
    │ 1 │ 1 │ 2 │    
┌───┼───┼───┴───┘    
│ 3 │ 3 │            
└───┴───┘

When the inner partition is $0$, the function numSemistandardTableaux computes the number of SSYT via the hook-content formula.

i3 : # allSemistandardTableaux(new Partition from {6,6,5,1,1})

o3 = 2520
 
i4 : numSemistandardTableaux(new Partition from {6,6,5,1,1})

o4 = 2520

See also

Ways to use allSemistandardTableaux:

  • allSemistandardTableaux(Partition)
  • allSemistandardTableaux(Partition,Partition)
  • allSemistandardTableaux(Partition,Partition,ZZ)
  • allSemistandardTableaux(Partition,ZZ)

For the programmer

The object allSemistandardTableaux is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Tableaux/documentation.m2:1146:0.