Sequences of ±1 that have partial sum >2 is a Catalan number We want to prove that the number of sequences $(a_1,...,a_{2n})$ such that
$$
• \text{ every } a_i \text{ is equal to} ±1;\\• a_1 + ··· + a_{2n} = 0;\\• \text{ every partial sum satisfies } a_1 + ··· + a_i > −2 
$$
is a Catalan number.
I've been trying to form a bijection between this and ballot sequences of size 2(n+1) by showing that you can remove any one +1 and any one -1 to yield our new sequences, but I am not sure if this is the best method. Any help would be super helpful!

This question is similar to the paths (0,2) to (2n,2) that do not touch X axis. The number of paths touches at least one point on the axis should be $2n \choose n+2$.
The total number of paths should be $2n \choose n$.
Then you can get the final answer, $2n \choose n+2$ - $2n \choose n$
