Probability of roll X or greater on Y Dice.

I want to make sure my analysis of the problem thus far is correct. I'm just learning about integer partitions so I am not sure exactly how to calculate the function I define as $\displaystyle G(x)$.

Assume that $\displaystyle P(x,y) $ is the probality of rolling exactly X on Y Dice.

This should be:

$\displaystyle P(x,y)=\frac{\binom{x-1}{y}-G(x)}{6^Y}$

where:

$\displaystyle G(x)$ is the number of partitions of X that include Y members and at least 1 member over 6.

The probability of rolling X or greater on Y dice is then:

$\displaystyle \sum_{i=x}^{6^y} P(x,y)$.

Calculating $\displaystyle G(x)$:

Define $\displaystyle part(n,k)$ to be the number of partitions of n each of whose member is $\displaystyle \geq k$. This is equivalent to the partitions of n which have at least k members.

$\displaystyle part(n,k) = 0\ for\ k>n$

$\displaystyle part(n,k) = 1\ for\ k=n$

$\displaystyle part(n,k) = part(n,k+1)+part(n-k,k)$

$\displaystyle part(n,1) =$ all the possible partitions of n

So

$\displaystyle part(x,y) -part(x,y-1)$ should be the all the ways I can partition X into exactly Y parts. However this could include members whose vlaue > 6.

(Also is the above just equal to $\displaystyle \binom{x-1}{Y}$

Or does the above imply that every member is also at least Y? (which is not at all what i want.)

I'm not sure at this point how to weed out the partitions that include invalid dice rolls?