避免对角模式的斜杨图填充
Fillings of skew shapes avoiding diagonal patterns
AI总结：
研究斜杨图的整数填充，证明 D-free 形状中避免东北链与东南链的填充双射，并构造任意斜杨图避免东南链与避免东北链及特定子填充的 01-填充双射。
摘要：
斜杨图是两个共享左上角的左上对齐 Ferrers 形状的差集。我们研究斜杨图的整数填充。作为我们的第一个主要结果，我们证明对于一类特定的斜杨图的遗传类，我们称之为 D-free 形状，避免大小为 $k$ 的东北链的填充与避免相同大小的东南链的填充是双射的。由于 Ferrers 形状是 D-free 形状的子类，这一结果可视为先前关于 Ferrers 形状的类似结果的推广。作为我们的第二个主要结果，我们在任意斜杨图的避免大小为 2 的东南链的 01-填充，与同一形状的同时避免大小为 2 的东北链和特定非方形子填充的 01-填充之间构造了一个双射。这推广了先前关于横向填充的结果。
英文摘要：
A skew shape is the difference of two top-left justified Ferrers shapes sharing the same top-left corner. We study integer fillings of skew shapes. As our first main result, we show that for a specific hereditary class of skew shapes, which we call D-free shapes, the fillings that avoid a north-east chain of size $k$ are in bijection with fillings that avoid a south-east chain of the same size. Since Ferrers shapes are a subclass of D-free shapes, this result can be seen as a generalization of previous analogous results for Ferrers shapes. As our second main result, we construct a bijection between 01-fillings of an arbitrary skew shape that avoid a south-east chain of size 2, and the 01-fillings of the same shape that simultaneously avoid a north-east chain of size 2 and a particular non-square subfilling. This generalizes a previous result for transversal fillings.
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