hookLength -- compute the hook length of a box of a tableau

Usage:

n = hookLength(thePosition,T)

m = hookLength lam
Inputs:
- thePosition, a sequence, a pair (i,j) of integers.
- T, an instance of the type YoungTableau, a tableau.
- lam, an instance of the type Partition, a partition.
Outputs:
- n, an integer, the hook length of box (i,j) in T.
- m, an integer, the hook length formula for lam.

Description

The hook length of box (i,j) is defined as the number of boxes directly below, and directly to the right, of a box, including the box itself.

i1 : T = youngTableau(new Partition from {6,6,5,3,1}, new Partition from {2,1,1})

             ┌───┬───┬───┬───┐
o1 =         │   │   │   │   │
         ┌───┼───┼───┼───┼───┤
         │   │   │   │   │   │
         ├───┼───┼───┼───┼───┘
         │   │   │   │   │    
     ┌───┼───┼───┼───┴───┘    
     │   │   │   │            
     ├───┼───┴───┘            
     │   │                    
     └───┘                    

o1 : YoungTableau

i2 : hookLength((1,1),T)

o2 = 7

i3 : applyPositions(T, thePosition -> hookLength(thePosition,T))

             ┌───┬───┬───┬───┐
o3 =         │ 7 │ 5 │ 4 │ 2 │
         ┌───┼───┼───┼───┼───┤
         │ 7 │ 6 │ 4 │ 3 │ 1 │
         ├───┼───┼───┼───┼───┘
         │ 5 │ 4 │ 2 │ 1 │    
     ┌───┼───┼───┼───┴───┘    
     │ 4 │ 2 │ 1 │            
     ├───┼───┴───┘            
     │ 1 │                    
     └───┘                    

o3 : YoungTableau

The hook length formula computes the number of standard Young tableau of shape $\lambda$.

i4 : lam = new Partition from {3,2,2}

o4 = Partition{3, 2, 2}

o4 : Partition

i5 : hookLength lam

o5 = 21

Ways to use hookLength:

hookLength(Partition)
hookLength(Sequence,YoungTableau)

For the programmer

The object hookLength is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Tableaux/documentation.m2:522:0.

hookLength -- compute the hook length of a box of a tableau

Description

See also

Ways to use hookLength:

For the programmer