How many Young diagrams are there with length of at most $p$ rows and $q$ columns, without restrictions on the weight of the diagrams?
The previous 2 questions asked for
Total number of Young diagrams with weight 6 and
Total number of Young diagrams with weight 7, excluding diagrams with more than 3 rows.
The answers I got are 1) 11 and 2) 8,
By counting the number of partitions of the integers 6 and 7 respectively, and subtracting $7, 6+1, 5+2, 5+1+1, 4+3, 4+2+1, 4+1+1+1$ from all partitions of 7.
However, I have no idea how to do that with unspecified weight, and taking into account the question details.
Notice the question is talking about Young diagrams, as opposed to Young tableaux.