Each element has a different number, which stipulates that it must be entered into the stack in ascending order of numbers. The order of leaving is not limited$.$ Each element must enter and exit the stack once, and the entire process must comply with the stacking last$-$in$-$first$-$out rule. The result can produce different permutation orders. $($a$)$ There are currently four elements, numbered $1,2,3,$ and $4$ respectively. If you want to use the above method to arrange them, what kind of arrangement orders are possible? $(10\%)($b$)$ If the question is changed to n elements, how many possible arrangements are there? $(5\%)$