FUNCTION: SfZee - compute the scalar product of a p-function with itself

CALLING SEQUENCE:

SfZee(part)

SfZee(part, q)

SfZee(part, q, t)

SYMF[SfZee](part)

SYMF[SfZee](part, q)

SYMF[SfZee](part, q, t)

PARAMETERS:

part = any list denoting a partition

q, t = any expressions

SYNOPSIS:

• Let part=[...3^i 2^j 1^k], and let sf denote the product ...p3^i*p2^j* p1^k, then SfZee(part) = SfScalar(sf, sf) = ... 3^i i! 2^j j! 1^k k!.

• The argument must be a partition, that is a weakly decreasing list of positive integers.

• When used with a second parameter q, it corresponds to a usual deformation of the standard SfScalar(sf, sf) scalar product. Thus: SfZee(part, q) = SfZee(part)*q^(nops(part)).

• When used with three paramaters, it gives another deformation of the usual scalar product: SfZee(part, q, t) = SfZee(part) multiplied by all (1-q^part[i])/(1-t^part[i]) for i from 1 to nops(part).

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

EXAMPLES:

> with(SYMF):
> SfZee([4,3,1,1]);
 
                                       24
 
> SfZee([4,3,1,1], 'q');
 
                                        4
                                    24 q
 
> SfZee([4,3,1,1], 'q', 't');
 
                                 4        3         2
                           (1 - q ) (1 - q ) (1 - q)
                        24 --------------------------
                                 4        3         2
                           (1 - t ) (1 - t ) (1 - t)