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. Nous démontrons que les remplissages maximaux avec 0 et 1 des polyominos $L$-convexes, avec longueurs de chaînes restreintes, peuvent être identifiés avec certains $\textit{rc-graphes}$, également connus sous le nom de $\textit{pipe dreams}$. En particulier, ceci montre un lien entre ces remplissages d'un diagramme de Ferrers et les polynômes de Schubert. On en déduit en outre une preuve bijective du fait que le nombre de remplissages maximaux d'un $\textit{stack polyomino}$ $S$, avec longueurs de chaînes bornées par un entier $k$, dépend seulement de $k$ et du multi-ensemble des tailles des colonnes de $S$. Notre contribution principale est un énoncé un peu plus fort, qui nous mène à conjecturer que l'ensemble ordonné (poset) des $\textit{rc-graphes}$ est en fait un treillis.
Highlights
1.1 Triangulations, multitriangulations and 0-1-fillingsThe systematic study of 0-1-fillings of polyominoes with restricted chain lengths likely originates in an article by Jakob Jonsson [5]
Vincent Pilaud and Michel Pocchiola [11] discovered rc-graphs more generally for multitriangulations, they were unaware of the theory of Schubert polynomials
Definition 2.1 A polyomino is a finite subset of the quarter plane N2, where we regard an element of N2 as a cell
Summary
The systematic study of 0-1-fillings of polyominoes with restricted chain lengths likely originates in an article by Jakob Jonsson [5] At first, he was interested in a generalisation of triangulations, where the objects under consideration are maximal sets of diagonals of the n-gon, such that at most k diagonals are allowed to cross mutually. Instead of studying fillings of the staircase shape only, he went on to consider more general shapes which he called stack and moon polyominoes, see Definition 2.2 and Figure 1 For stack polyominoes he was able to prove that the number of maximal fillings depends only on k and the multiset of heights of the columns, not on the particular shape of the polyomino.
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