In a textbook there is a question about partitions that follows:
Find the number of partitions of 8 that have
(c) at most three positive parts;
(d) exactly three nonnegative parts.
For c) I simply got $P(8,1)$ + $P(8,2)$ + $P(8,3)$ = $10$
and for d) I got the same answer by doing P(8,3) plus all of the partitions that can have $0$s in them ${ (008), (017), (026), (035), (044) }$ So $5 + 5 = 10$
The next question in the book asks to find $P(11,3)$ which equals $10$ as well and then asks to explain why parts c), d) and $P(11.3)$ are equal. I am having a difficult time understanding how all these relate to each other. I am trying to create a practical example in my head but no luck. If anyone could help I would really appreciate it.
For c and d, you have essentially described the natural bijection: pad with $0$ if necessary.
For a bijection between d and $P(11,3)$: given a partition of $8$ into exactly three nonnegative parts, add $1$ to each part; given a partition of $11$ into exactly $3$ parts, subtract $1$ from each part.