MathDB

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)

6

, and these lists are

(4),

(1,3),

(2,2),

(1,1,2),

(2,1,1),

(1,1,1,1)

Problem Statement