toPartitionChain -- decompose a tableau into a chain of partitions

Usage:

toPartitionChain T
Inputs:
- T, an instance of the type YoungTableau, a tableau.
Outputs:
- s, a sequence, a sequence of partitions.

Description

A (semistandard) Young tableau of shape $\lambda/\mu$ determines (and is determined by) a sequence of partitions $(\lambda^{(0)},\ldots,\lambda^{(r)})$, where $\mu= \lambda^{(0)}\subseteq\lambda^{(1)}\subseteq\cdots\subseteq\lambda^{(r)}=\lambda$.

i1 : lam = new Partition from {6,6,5,3,2,2,2,1}

o1 = Partition{6, 6, 5, 3, 2, 2, 2, 1}

o1 : Partition

i2 : mu = new Partition from {3,1}

o2 = Partition{3, 1}

o2 : Partition

i3 : entryList = {1,1,7,1,3,4,4,8,1,2,4,7,7,2,3,5,3,4,4,5,5,6,8}

o3 = {1, 1, 7, 1, 3, 4, 4, 8, 1, 2, 4, 7, 7, 2, 3, 5, 3, 4, 4, 5, 5, 6, 8}

o3 : List

i4 : T = youngTableau(lam,mu,entryList)

                 ┌───┬───┬───┐
o4 =             │ 1 │ 1 │ 7 │
         ┌───┬───┼───┼───┼───┤
         │ 1 │ 3 │ 4 │ 4 │ 8 │
     ┌───┼───┼───┼───┼───┼───┘
     │ 1 │ 2 │ 4 │ 7 │ 7 │    
     ├───┼───┼───┼───┴───┘    
     │ 2 │ 3 │ 5 │            
     ├───┼───┼───┘            
     │ 3 │ 4 │                
     ├───┼───┤                
     │ 4 │ 5 │                
     ├───┼───┤                
     │ 5 │ 6 │                
     ├───┼───┘                
     │ 8 │                    
     └───┘                    

o4 : YoungTableau

i5 : theDecomp = toPartitionChain T

o5 = (Partition{3, 1, 0, 0, 0, 0, 0, 0}, Partition{5, 2, 1, 0, 0, 0, 0, 0},
     ------------------------------------------------------------------------
     Partition{5, 2, 2, 1, 0, 0, 0, 0}, Partition{5, 3, 2, 2, 1, 0, 0, 0},
     ------------------------------------------------------------------------
     Partition{5, 5, 3, 2, 2, 1, 0, 0}, Partition{5, 5, 3, 3, 2, 2, 1, 0},
     ------------------------------------------------------------------------
     Partition{5, 5, 3, 3, 2, 2, 2, 0}, Partition{6, 5, 5, 3, 2, 2, 2, 0},
     ------------------------------------------------------------------------
     Partition{6, 6, 5, 3, 2, 2, 2, 1})

o5 : Sequence

This chain has the property that each $\lambda^{(i)}/\lambda^{(i-1)}$ is a horizontal strip.

i6 : drawInnerShape true 

o6 = true

i7 : for i from 1 to #theDecomp-1 do (
         print(toString(toSequence trim theDecomp#i)|"/"|toString(toSequence trim theDecomp#(i-1))|":");
         print youngTableau(theDecomp#i,theDecomp#(i-1));
         )
(5,2,1)/(3,1):
            ┌───┬───┐
  ■   ■   ■ │   │   │
    ┌───┐   └───┴───┘
  ■ │   │            
┌───┼───┘            
│   │                
└───┘                
(5,2,2,1)/(5,2,1):
                     
  ■   ■   ■   ■   ■  
                     
  ■   ■              
    ┌───┐            
  ■ │   │            
┌───┼───┘            
│   │                
└───┘                
(5,3,2,2,1)/(5,2,2,1):
                     
  ■   ■   ■   ■   ■  
        ┌───┐        
  ■   ■ │   │        
        └───┘        
  ■   ■              
    ┌───┐            
  ■ │   │            
┌───┼───┘            
│   │                
└───┘                
(5,5,3,2,2,1)/(5,3,2,2,1):
                     
  ■   ■   ■   ■   ■  
            ┌───┬───┐
  ■   ■   ■ │   │   │
        ┌───┼───┴───┘
  ■   ■ │   │        
        └───┘        
  ■   ■              
    ┌───┐            
  ■ │   │            
┌───┼───┘            
│   │                
└───┘                
(5,5,3,3,2,2,1)/(5,5,3,2,2,1):
                     
  ■   ■   ■   ■   ■  
                     
  ■   ■   ■   ■   ■  
                     
  ■   ■   ■          
        ┌───┐        
  ■   ■ │   │        
        └───┘        
  ■   ■              
    ┌───┐            
  ■ │   │            
┌───┼───┘            
│   │                
└───┘                
(5,5,3,3,2,2,2)/(5,5,3,3,2,2,1):
                     
  ■   ■   ■   ■   ■  
                     
  ■   ■   ■   ■   ■  
                     
  ■   ■   ■          
                     
  ■   ■   ■          
                     
  ■   ■              
                     
  ■   ■              
    ┌───┐            
  ■ │   │            
    └───┘            
(6,5,5,3,2,2,2)/(5,5,3,3,2,2,2):
                    ┌───┐
  ■   ■   ■   ■   ■ │   │
                    └───┘
  ■   ■   ■   ■   ■      
            ┌───┬───┐    
  ■   ■   ■ │   │   │    
            └───┴───┘    
  ■   ■   ■              
                         
  ■   ■                  
                         
  ■   ■                  
                         
  ■   ■                  
                         
(6,6,5,3,2,2,2,1)/(6,5,5,3,2,2,2):
                         
  ■   ■   ■   ■   ■   ■  
                    ┌───┐
  ■   ■   ■   ■   ■ │   │
                    └───┘
  ■   ■   ■   ■   ■      
                         
  ■   ■   ■              
                         
  ■   ■                  
                         
  ■   ■                  
                         
  ■   ■                  
┌───┐                    
│   │                    
└───┘

Ways to use toPartitionChain:

toPartitionChain(YoungTableau)

For the programmer

The object toPartitionChain is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Tableaux/documentation.m2:423:0.