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!
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Hint. Catalan numbers $C_n$ can be used to count the number of "monotonic lattice paths along the edges of a grid with $n \times n$ square cells, which do not pass above the diagonal. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards" (quoted from Wikipedia). Here you have the case $n=4$.
In order to describe the movements, let $a_i$ be the $i$-th movement and choose $a_i=1$ if you move to the right and $a_i=-1$ if you move upwards. Is it clear why $\sum a_i=0$? And why every $a_i$ is equal to $\pm 1$? Why do they satisfy $a_1+a_2+\ldots+ a_k<-2$ (look at the diagonal)?
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, $2n \choose n+2$ - $2n \choose n$