Given positive integers $m,n(2\le m\le n)$, let $a_1,a_2,\ldots ,a_m$ be a permutation of any $m$ pairwise distinct numbers taken from $1,2,\ldots ,n$. If there exist $k\in\{1,2,\ldots ,m\}$ such that $a_k+k$ is odd, or there exist positive integers $k,l(1\le k<l\le m)$ such that $a_k>a_l$, then call $a_1,a_2,\ldots ,a_m$ a [i]good[/i] sequence. Find the number of good sequences.
1. **Define the problem in terms of permutations:**
   We need to find the number of permutations of $m$ distinct numbers taken from $\{1, 2, \ldots, n\}$ that are "good" sequences. A sequence is "good" if:
   - There exists $k \in \{1, 2, \ldots, m\}$ such that $a_k + k$ is odd, or
   - There exist $k, l$ with $1 \le k < l \le m$ such that $a_k > a_l$.

2. **Consider the total number of permutations:**
   The total number of permutations of $m$ distinct numbers from $\{1, 2, \ldots, n\}$ is given by:
   \[
   P(n, m) = \frac{n!}{(n-m)!}
   \]

3. **Define the "bad" sequences:**
   A sequence is "bad" if:
   - For all $k \in \{1, 2, \ldots, m\}$, $a_k + k$ is even, and
   - For all $1 \le k < l \le m$, $a_k < a_l$ (i.e., the sequence is strictly increasing).

4. **Determine the conditions for a "bad" sequence:**
   - For $a_k + k$ to be even, $a_k$ and $k$ must both be even or both be odd.
   - Since $a_1, a_2, \ldots, a_m$ must be strictly increasing, the sequence must be an increasing subsequence of $\{1, 2, \ldots, n\}$.

5. **Count the number of "bad" sequences:**
   - We need to choose $m$ numbers from $\{1, 2, \ldots, n\}$ such that they form an increasing sequence and satisfy the parity condition.
   - There are $\binom{n}{m}$ ways to choose $m$ numbers from $n$ numbers.
   - For each chosen subset, we need to check if the parity condition is satisfied. This requires that the positions and values align in parity (both even or both odd).

6. **Calculate the number of "bad" sequences:**
   - For each $k$, $a_k$ must be chosen such that $a_k \equiv k \pmod{2}$.
   - There are $\left\lfloor \frac{n}{2} \right\rfloor$ even numbers and $\left\lceil \frac{n}{2} \right\rceil$ odd numbers in $\{1, 2, \ldots, n\}$.
   - We need to choose $m$ numbers such that half of them are even and half are odd (or as close as possible).

7. **Subtract the number of "bad" sequences from the total number of permutations:**
   - The number of "good" sequences is given by:
   \[
   \text{Number of good sequences} = P(n, m) - \text{Number of bad sequences}
   \]

The final answer is $\boxed{P(n, m) - \text{Number of bad sequences}}$