There is a multiset $\{\frac{n-1}{2} a, \frac{n+1}{2} b\}$ where $n$ is an odd number. How many permutations of this multiset have the property that reading from left to right $b$ exceeds $a$ only for the complete permutation?
For $n=5$ I get $a, b, a, b, b$ and $a, a, b, b, b$ so the answer is 2.
In other words: I draw red and blue balls. How many permutations of $n$ draws have the property that the number of blue balls exceeds the number of red balls only after the $n$'th draw?
I am looking for a general result. I calculated the result for $n=1,3,...,13,15$ and found the sequence 1, 1, 2, 5, 14, 42, 142, 552. OEIS does not know the sequence. Unfortunately my sequence does not consist of the Catalan Numbers.
The number of possible combinations of a and b that lead to b exceeding a after the $n$'th draw (for the first time) is given by the catalan number $C_{\frac{n-1}{2}}={n-1 \choose (n+1)/2}-{n-1 \choose (n+1)/2}$. This is because the last draw has to be b and then the problem is equivalent to finding the restricted number of paths as cited in the comments.