Our main contribution here is the discovery of a new family of standard Young tableaux $ {\cal T}^k_n$ which are in bijection with the family ${\cal D}_{m,n}$ of Rational Dyck paths for $m=k\times n\pm 1$ (the so called "Fuss" case). Using this family we give a new proof of the invertibility of the sweep map in the Fuss case by means of a very simple explicit algorithm. This new algorithm has running time $O(m+n)$. It is independent of the Thomas-William algorithm.

中文翻译：

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关于大惊小怪的有理 Dyck 路径的扫描图

我们在这里的主要贡献是发现了一个新的标准 Young tableaux 族 ${\cal T}^k_n$，它们与 ${\cal D}_{m,n}$ 的 Rational Dyck 路径族双射m=k\times n\pm 1$（所谓的“大惊小怪”的情况）。使用这个族，我们通过一个非常简单的显式算法给出了 Fuss 情况下扫描图可逆性的新证明。这个新算法的运行时间为 $O(m+n)$。它独立于 Thomas-William 算法。