FUNCTION: Part2ListHook - calculate the list of hook lengths

CALLING SEQUENCE:

Part2ListHook(part)

PART[Part2ListHook](part)

PARAMETERS:

part = any list denoting a partition

SYNOPSIS:

• The Part2ListHook function computes the list of all hook lengths of a given partition. The result is sorted into decreasing order.

• The arm and leg length of a cell (i,j) are respectively part[i] - j and cpart[j] - i where cpart is the conjugate of part.

• The call Part2ListHook(part,a) computes the Jack-hook-product. a may be any expression. This is the product: a*arm_length[i,j] + leg_length[i,j] + 1 over all cells of the diagram of the partition part.

• The call Part2ListHook(part, q, t) computes the two-variable hook polymial. Arguments q and t can be any expressions. This is the product: 1 - q^arm_length[i,j]*t^(1+leg_length[i,j]) over all cells of the diagram of the partition part.

• Whenever there is a conflict between the function name Part2ListHook and another name used in the same session, use the long form PART['Part2ListHook'].

EXAMPLES:

> with(PART):
> Part2ListHook([3,1]);
 
                                  [4, 2, 1, 1]
 
> 4!/convert([4, 2, 1, 1], `*`);
 
                                       3
 
> Part2ListHook([4,1,1], 'z');
 
                          2 (3 z + 3) (2 z + 1) (z + 1)
 
> Part2ListHook([4,1,1], 'q', 't');
 
                    3  3        2                     2       2
              (1 - q  t ) (1 - q  t) (1 - q t) (1 - t)  (1 - t )