MathDB

\forall n \in \mathbb{N}

P(n)

denotes the number of the partition of

n

as the sum of positive integers (disregarding the order of the parts), e.g. since

4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4

, so

P(4)=5

. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote

q(n)

is the sum of all the dispersions, e.g.

q(4)=1+2+2+1+1=7

n \geq 1

. Prove that (1)

q(n) = 1 + \sum^{n-1}_{i=1} P(i).

(2)

1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)

Problem Statement