Abstract
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.
Highlights
The systematic study of matchings and set partitions with certain restrictions on their crossings and nestings started in [2], where Chen et al used Robinson-Schensted-like insertion/deletion processes to show the symmetry between the sizes of the largest crossings and the largest nestings
By the problem of counting generalized triangulations with a given size of the maximal crossings, proved that the number of 01-fillings of a stack polyomino that do not contain a northeast chain of size k depends only on the distribution of lengths of the columns of the polyomino
Rubey proved that the number of 01-fillings with the longest northeast chains of size k and exactly ci non-zero entries in column i are equal for M and σM
Summary
The systematic study of matchings and set partitions with certain restrictions on their crossings and nestings started in [2], where Chen et al used Robinson-Schensted-like insertion/deletion processes to show the symmetry between the sizes of the largest crossings and the largest nestings. Rubey proved that the number of 01-fillings with the longest northeast chains of size k and exactly ci non-zero entries in column i are equal for M and σM. It is a very interesting property for fillings of moon polyominoes: many combinatorial statistics are invariant under permutations of rows (or columns).
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