MathDB

Let

n

be a positive integer. We consider the set

P

of all partitions of

n

into a sum of positive integers (the order is irrelevant). For every partition

\alpha

, let

a_{k}(\alpha)

be the number of summands in

\alpha

that are equal to

k, k = 1,2,...,n.

Prove that

\sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.

Problem Statement