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 .

Assume that is the probality of rolling exactly X on Y Dice.

This should be:

where:

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:

.

Calculating :

Define to be the number of partitions of n each of whose member is . This is equivalent to the partitions of n which have at least k members.

all the possible partitions of n

So

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

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?