Minimal Bridges and a Rotation-Based Bijection

A classical problem in lattice path enumeration counts paths that remain on one side of a boundary line. We study several classes of paths where this boundary is porous and show that they are related through a single half-turn rotation bijection. As a first application, we enumerate minimal bridges by relating them to excursions: for positive integers $k$ and $n$, the number of paths from $(0,0)$ to $(kn,n)$ with unit right and up steps that avoid all other lattice points on the line $y=x/k$ is $\frac{k}{kn+n-1}\binom{kn+n}{n}$. The same bijection yields a relation between the ordinary generating functions for binomial coefficients and $k$-Catalan numbers through a dual edge-forbidden model, extends to forbidden strips containing the diagonal, and handles a rational-slope case involving Duchon paths. Finally, our bijection also proves that the number of bridges from $(0,0)$ to $(2n,2n)$ that avoid even diagonal points is $C_{2n}+4C_{2n-1}$, with $C_n$ the $n$th Catalan number. This complements a result of Shapiro.