Parallelogram polyominoes and rectangular EW-tableaux: correspondences through the sandpile model

TL;DR

This paper establishes a direct bijection between rectangular EW-tableaux and labelled parallelogram polyominoes, simplifying the connection between these combinatorial objects and the sandpile model on bipartite graphs.

Contribution

It introduces marked rectangular EW-tableaux to encode all recurrent configurations, enabling a straightforward bijection with labelled parallelogram polyominoes, removing the need for recurrent configurations.

Findings

Bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes

Marked rectangular EW-tableaux encode all recurrent configurations

Enhanced understanding of the relationship between EW-tableaux and parallelogram polyominoes

Abstract

This paper establishes connections between EW-tableaux and parallelogram polyominoes by using recent research regarding the sandpile model on the complete bipartite graph. This paper presents and proves a direct bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes. The significance of this is that allows one to move between these objects without the need for `recurrent configurations', the central object which previously tied this work together. It introduces the notion of a marked rectangular EW-tableaux that exactly encode all recurrent configurations of the sandpile model on the complete bipartite graph. This shows how non-cornersupport entries that featured in previous work can be utilized in a simple but important way in relation to EW-tableaux. It lifts the bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes to a bijection between marked rectangular EW-tableaux and labelled parallelogram polyominoes. This bijection helps us to fully understand the aspects of these very different objects that are, in a sense, different sides of the same coin.