There are N objects of type A. There also N objects of type B. Consider random permutations of these 2N objects in to an array of size 2N. Let's call this array $A$. What is the number of permutations of these 2N objects such that when looking at the first m elements of $A$, for all $m$, the number of B elements is always $\leq$ the number of $A$ elements?
I can also write this out mathematically. Consider $N$ numbers of value $+1$ and $N$ numbers of value $-1$. These 2N numbers are stored in an array. What is number of permutations such that (assume array is 1-based)
$$ \sum_{i=1}^m A[i] \geq 0 \ \ \forall m \in \{1, \ldots, 2N\} $$
Note that the probability of a permutation adhering to the above constraint is $\frac{1}{n+1}$. This problem can be solved using the reflection principle, but I am looking for other approach to solving this problem.
Without thinking about the reflection principle, it's rather difficult for me to approach this problem in terms of permutations/combinations.