Let $P(k, n)$ stand for the number of partitions of $k$ into $n$ positive parts.
Find the number of partitions of $8$ that have
(a) any number of positive parts;
(b) at least three positive parts;
(c) at most three positive parts;
(d) exactly three nonnegative parts.
(a) $P(8, 1) + P(8, 2) + P(8, 3) + P(8, 4) + p(8, 5) + P(8, 6) + P(8, 7) + P(8, 8) $
(b) $P(8, 3) + P(8, 4) + p(8, 5) + P(8, 6) + P(8, 7) + P(8, 8) $
(c) $P(8, 1) + P(8, 2) + P(8, 3)$
(d) $P(8, 1) + P(8, 2) + P(8, 3)$
Can someone, please, confirm the answers in/correct? I need these to prove a recurrence relation so I'd like to have correct assumptions. Thanks.
That all looks right. The use of nonnegative in d) quite tricky, because P(k,n) counts positive parts, so the presence of 0s is allowed in d) even though P(k,n) does not include them as parts. This all seems correct to me.