Maximal increasing sequences in fillings of almost-moon polyominoes
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abstract
2015 Elsevier Inc. All rights reserved. It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino that do not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey's proof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces the result for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijective proof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removing one of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chain of the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sum of the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, the row sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijective proof of Rubey's result but also two refinements of it.
