极小桥与基于旋转的双射
Minimal Bridges and a Rotation-Based Bijection
AI总结：
该论文研究边界多孔的格路计数，通过半旋转双射关联多类路径，完成极小桥等格路的计数，补充了Shapiro的相关结果。
AI中文摘要：
格路计数中的经典问题是统计始终位于边界线一侧的路径。我们研究几类边界具有“多孔性”的路径，证明它们通过一个半旋转双射相关联。首个应用是将极小桥与游走路径关联以完成计数：对正整数k和n，从(0,0)到(kn,n)、步长为单位右步和上步且避开直线y=x/k上所有其他格点的路径数为$\frac{k}{kn+n-1}\binom{kn+n}{n}$。该双射还通过对偶禁边模型建立了二项式系数与k-Catalan数普通生成函数的关系，可推广至含对角线的禁带情形，并处理涉及Duchon路径的有理斜率情况。最后，该双射还证明从(0,0)到(2n,2n)、避开偶对角点的桥的数量为$C_{2n}+4C_{2n-1}$（其中$C_n$为第n个Catalan数），这补充了Shapiro的一项结果。
英文摘要：
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.
