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

Usage:

allSemistandardTableaux(lam,mu,N)
Inputs:
- lam, an instance of the type Partition, the outer shape, $\lambda$.
- mu, an instance of the type Partition, the inner shape, $\mu$.
- N, an integer, the maximum entry.
Outputs:
- b, a bag, bagged list of all SSYT of given shape and entries in 1..N. If N is not provided, it is assumed to be the length of the 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

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 Tableaux/documentation.m2:1146:0.

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

Description

See also

Ways to use allSemistandardTableaux:

For the programmer