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Maximal 0-1-fillings of moon polyominoes with restricted chain lengths and rc-graphs

Abstract : We show that maximal 0-1-fillings of moon polynomials, with restricted chain lengths, can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0-1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino $S$ with no north-east chains longer than $k$ depends only on $k$ and the multiset of column heights of $S$. Our main contribution is a slightly stronger theorem, which in turn leads us to conjecture that the poset of rc-graphs with covering relation given by generalised chute moves is in fact a lattice.
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Martin Rubey. Maximal 0-1-fillings of moon polyominoes with restricted chain lengths and rc-graphs. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.835-848. ⟨hal-01215066⟩

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