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

TL;DR

This paper establishes a connection between maximal 0-1 fillings of moon polyominoes with chain-length restrictions and rc-graphs, revealing new combinatorial insights and conjecturing a lattice structure for rc-graphs.

Contribution

It introduces a bijective correspondence between maximal fillings and rc-graphs, and proves a dependence of the count on chain-length and column heights, along with a conjecture on the poset structure.

Findings

Maximal fillings correspond to specific rc-graphs.

Number of fillings depends only on chain-length and column multiset.

Conjecture that rc-graphs form a lattice under chute moves.

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.