Prove : $P(n | \text{ number of parts $\le m$}) = P(n | \text{ all parts $\le m$})$

Question

I'm trying to prove both sides of :

$$P(n | \text{ number of parts $\le m$}) = P(n | \text{ all parts $\le m$}).$$

First side: Given a partition where all parts $\le m$, we can build a Ferrer's graph where

$$\lambda_1 \ge \lambda_2 > \lambda_3 > \dots > \lambda_k.$$

λ1 .............
λ2 .............
.
.
.
λk .............

and then conjugate the graph, and we'd get that there are at most $m$ parts of $\lambda_i$.

But how can I do the other side:

Given a partition where number parts $\le m$, we want to prove that all parts are $\le m$?

Thanks

Answer 1

I think you can set up a 1-1 correspondence between these types of partitions by drawing a Ferrer's graph for a partition with the number of parts $\le m$, and then observing that the conjugate Ferrer's graph (ie, viewing the diagram vertically) gives a partition with all parts $\le m$.