= Reflection count for Dyck paths
There are $\binom{2n}{n}$ paths with $n$ up steps and $n$ down steps, ignoring the height constraint. For $n\geq1$, reflect the prefix of a bad path through its first visit to height $-1$, swapping its up and down steps. The resulting path has $n+1$ up steps and $n-1$ down steps, hence final height two. Its first visit to height one identifies the inverse reflection, so this is a <bijection>. There are $\binom{2n}{n+1}$ bad paths. Subtracting gives
$$
C_n=\binom{2n}{n}-\binom{2n}{n+1}=\frac1{n+1}\binom{2n}{n}.
$$
This count connects a global nonnegative-height requirement to an unrestricted <binomial coefficient>. The case $n=0$ is separately the unique empty path.
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