Two New Bijections on Lattice Paths

Suppose \( 2n \) voters vote sequentially for one of two candidates. For how many such sequences does one candidate have strictly more votes than the other at each stage of the voting? The answer is \( \binom{2n}{n} \) and, while easy enough to prove using generating functions, for example, only three combinatorial proofs exist, due to Kleitman, Gessel, and Callan. In this paper, we present two new bijective proofs.