CGMO 2020 Problem 8

Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers.

Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that:(1) $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$ .(2) The number of all pairs of integers $(i,j)$ satisfying $1\leq i<j\leq m$ and $a_i>a_j$ is even.

For example, when $n=4$ , the number of all such lists $(a_1,a_2,\cdots,a_m)$ is $6$ , and these lists are $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$ .