Assume there is a unsatisfied permutation $$$P$$$ and $$$x$$$ is the number of $$$i$$$ that satisfy $$$p_i < p_{i + 1}$$$ and $$$x < n$$$. So just reverse $$$P$$$, the number of $$$i$$$ satisfy $$$p_i < p_{i + 1}$$$ in our new permutation is $$$2 * n - 1 - x > 2 * n - n - 1 = n - 1$$$. It's easy to see that the answer is $$$\frac{(2*n)!}{2}$$$

if any permutation doesn't satisfied that condition then it reverse will do and if that permutation does then its reverse doesn't.

so half will satisfy and other half( reverse one) will not.