Abstract
It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it. Rubey a montré que le nombre de remplissages d’un polyomino lunaire donné par des zéros et des uns quine contiennent pas de chaîne nord-est d’une taille fixée ne dépend que de l’ensemble des longueurs des colonnesdu polyomino. La preuve de Rubey utilise une adaptation du jeu de taquin et de la promotion sur des remplissagesarbitraires de polyominos lunaires et déduit le résultat pour les remplissages 0/1 par inclusion-exclusion. Dans cetarticle, nous présentons la première preuve bijective de ce résultat en considérant des remplissages de polyominospresque lunaires, qui sont des polyominos lunaires dont on a supprimé une ligne. Plus précisément, nous construisonsune bijection simple qui préserve la taille de la plus longue chaîne nord-est des remplissages lorsque deux lignesadjacentes du polyomino sont échangées. Cette bijection préserve aussi la somme des colonnes des remplissages. Enoutre, nous présentons aussi une bijection simple qui préserve la taille de la plus longue chaîne nord-est, la sommedes lignes et la somme des colonnes si chaque ligne du remplissage contient au plus un 1. Nous fournissons donc nonseulement une preuve bijective du résultat de Rubey, mais aussi deux raffinements de celui-ci.
Highlights
The motivation for this paper is the very important result of Rubey [18] that the cardinalities of the sets {M : M ∈ FN(M, n) and ne(M ) = k} and {M : M ∈ F01(M, n) and ne(M ) = k} only depend on the multiset of column lengths of the moon polyomino M. This is a very interesting property for fillings of moon polyominoes: many combinatorial statistics are invariant under permutations of rows
A similar statement is known for the major index introduced by Chen, Poznanovic, Yan and Yang [5], for the numbers of northeast and southeast chains of length 2 studied by Kasraoui [14], and for various analogs and generalizations of 2-chains [6, 21]
Our first main result states that if M and N are two almost-moon polyominoes related by an interchange of adjacent rows (e.g. Figure 1), the statistic ne is equidistributed over the sets F(M, ∗, c) and F(N, ∗, c) of fillings of M and N, respectively, with fixed column sums but arbitrary row sums: qne(M) =
Summary
In [4], Chen, Deng, Du, Stanley, and Yan used Robinson-Schensted-like insertion/deletion processes to show the symmetry between the sizes of the largest crossings and the largest nestings in set partitions. The motivation for this paper is the very important result of Rubey [18] that the cardinalities of the sets {M : M ∈ FN(M, n) and ne(M ) = k} and {M : M ∈ F01(M, n) and ne(M ) = k} only depend on the multiset of column lengths of the moon polyomino M. The second map ψM,N is restricted to fillings in which every row has at most one 1 and preserves the size of the longest northeast chains, the row sum, and the column sum Both our maps are simple to describe but the proofs that they have the desired properties are technical; we omit them here and refer the reader to the full version of the paper [17].
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