The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order $k$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most $(k-1)$ many, and this bound is sharp. We use this to give a simple proof of the “permutation removal lemma.”

中文翻译：

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关于模式避免排列

排列极限理论引出了称为排列的极限对象，它是单位平方上的某些 Borel 测度。我们证明排列可以避免给定的顺序排列$k$具有特别简单的结构。也就是说，几乎每个排列分解的纤维（例如，沿 x 轴）最多仅由原子组成$（k-1）$很多，而且这个界限很尖锐。我们用它来给出“排列移除引理”的简单证明。