For each partition $\sigma = (\lambda_1,\ldots,\lambda_k)$, define the weight function $w^∗(σ) = k$. Let $\Phi^∗P_n (x)$ be the generating function for $P_n$ with respect to $w^*$. Prove that for all $n\in\Bbb N$, $\Phi_{P_n} =\Phi^∗_{P_n}$, where $\Phi_{P_n}(x)$ is the generating function for $P_n$ with weight function $w(\sigma)=\lambda_1$.

(A partition of $n$ is a monotone decreasing sequence of positive integers which sum up to $n$; i.e. $(\lambda_1,\ldots,\lambda_k)$ where $\lambda_1 +\cdots+\lambda_k = n$ and $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k > 0$. Let $P_n$ be the set of all partitions of $n$.)

How to prove this please help.


HINT: You need either to say what $\Phi_{P_n}$ is or include a link to your earlier question; I added the missing definition to your question.

The most straightforward way to show that ${\Phi}^*_{P_n}=\Phi_{P_n}$ is to show that for each $n$ and $k$, the number of partitions of $n$ with $k$ parts is the same as the number of partitions of $n$ with largest part $k$. (Why?) This is easily proved by looking at the Ferrers diagrams of conjugate partitions.


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