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:
[b](1)[/b] $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$.
[b](2)[/b] 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 $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$.
1. **Define the problem and notation:**
   - Let \( n \) be a given positive integer.
   - Let \(\mathbb{N}_+\) denote the set of all positive integers.
   - We need to determine the number of all finite lists \((a_1, a_2, \ldots, a_m)\) such that:
     1. \( m \in \mathbb{N}_+ \) and \( a_1, a_2, \ldots, a_m \in \mathbb{N}_+ \) 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.

2. **Define compositions and inversion number:**
   - A finite list \((a_1, a_2, \ldots, a_m)\) satisfying condition (1) is called a composition of \( n \). Let \( C \) denote the set of all compositions of \( n \).
   - The number of all pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq m\) and \(a_i > a_j\) is called the inversion number of \((a_1, a_2, \ldots, a_m)\).

3. **Define sets \( A \) and \( B \):**
   - Let \( A \) be the set of all compositions of \( n \) having an even inversion number.
   - Let \( B \) be the set of all compositions of \( n \) having an odd inversion number.
   - We need to prove that \(|A| = 2^{n-2} + 2^{\left\lfloor n/2 \right\rfloor - 1}\).

4. **Count the total number of compositions:**
   - The total number of compositions of \( n \) is \( 2^{n-1} \).

5. **Establish the relationship between \( |A| \) and \( |B| \):**
   - Since \( A \) and \( B \) are disjoint and their union is \( C \), we have:
     \[
     |A| + |B| = |C| = 2^{n-1}
     \]

6. **Find the difference \( |A| - |B| \):**
   - We need to show that \( |A| - |B| = 2^{\left\lfloor n/2 \right\rfloor} \).

7. **Define the matching (involution) \(\iota\):**
   - For \((a_1, a_2, \ldots, a_m) \in B\), there exists some \( i \leq m/2 \) such that \( a_{2i-1} \neq a_{2i} \).
   - Swap \( a_{2i-1} \) and \( a_{2i} \) to get a new composition in \( A \).
   - Define \(\iota(a_1, a_2, \ldots, a_m)\) as this new composition.

8. **Define the set \( A' \):**
   - Let \( A' \) be the set of all compositions \((a_1, a_2, \ldots, a_m)\) of \( n \) that satisfy \( a_1 = a_2 \), \( a_3 = a_4 \), etc.
   - \( A' \subseteq A \).

9. **Construct the inverse map \(\kappa\):**
   - Define \(\kappa: A \setminus A' \to B\) similarly to \(\iota\).
   - \(\kappa\) is a bijection, so \( |B| = |A \setminus A'| \).

10. **Calculate \( |A'| \):**
    - Show that \( |A'| = 2^{\left\lfloor n/2 \right\rfloor} \) by induction on \( n \).

11. **Solve the system of equations:**
    - We have:
      \[
      |A| + |B| = 2^{n-1}
      \]
      \[
      |A| - |B| = 2^{\left\lfloor n/2 \right\rfloor}
      \]
    - Adding these equations:
      \[
      2|A| = 2^{n-1} + 2^{\left\lfloor n/2 \right\rfloor}
      \]
      \[
      |A| = \frac{2^{n-1} + 2^{\left\lfloor n/2 \right\rfloor}}{2} = 2^{n-2} + 2^{\left\lfloor n/2 \right\rfloor - 1}
      \]

The final answer is \( \boxed{ 2^{n-2} + 2^{\left\lfloor n/2 \right\rfloor - 1} } \).