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arXiv · 2002.12308

Fillings of skew shapes avoiding diagonal patterns

Vít Jelínek·Mark Karpilovskij
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Abstract

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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Vít Jelínek, Mark Karpilovskij. 2020-02-27. Fillings of skew shapes avoiding diagonal patterns. https://doi.org/10.46298/dmtcs.6171

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