Relation between partitions of $n$ into $k$ distinct parts and partitions of $n$ into at most $k$ parts

Asked Dec 12 '18 at 23:36

Active Dec 12 '18 at 23:36

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I'm working on a problem that I'm completely stuck on: Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct (unequal) parts. Prove that the number of partitions of $n$ into at most $k$ parts (parts can be equal here) is $Q(n + \binom{k+1}{2} , k)$

Any hints?

asked Dec 12 '18 at 23:36

hopelessundergrad

Consider the conjugate of the partition and take $k$ of the first one, take $k-1$ of the second one and take $1$ for the last one for a total of $1+2+\cdots +k=\binom{k+1}{2}$ – Phicar Dec 12 '18 at 23:38
What is a conjugate of a partition? – user9077 Dec 13 '18 at 00:06
4

By searching for distinct at most parts I quickly found Combinatorics problem based on Ferrers graph, which is a stronger form of this question. – Peter Taylor Dec 13 '18 at 11:32
This is a problem that has cropped up here previously, but the OP does not necessarily know about Ferrers diagrams or the "boxy" alternative Young tableaux. The only Answer on the proposed target duplicate is pretty terse and assumes some familiarity of that kind. I'm going to suggest looking for another target that includes an illustration for the sake of clarity. – hardmath Dec 13 '18 at 16:30
Here's another good explanation, just with words and no pictures. I will try and add a "picture" to the target duplicate above, for clarity. – hardmath Dec 13 '18 at 17:00

Relation between partitions of $n$ into $k$ distinct parts and partitions of $n$ into at most $k$ parts

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