Which permutations can not be obtained by moving elements through two stacks?
This process is called stack sorting. Wikipedia has a page on the subject. 

Here you read that the number of possible sequences when the input is $1,2,\dots,n$ equals the Catalan number $C_n = \frac{1}{n+1} {2n\choose n}$, which differs from your guess.

In fact there is a nice characterization of the orders which are possible, they avoid the permutation pattern $231$. However your problem is "backwards", not the strings that can be sorted using a single stack, but the strings that can be obtained that way.