Increasing and Decreasing Sequences in Fillings of Moon Polyominoes

TL;DR

This paper adapts jeu de taquin and promotion techniques to study fillings of moon polyominoes, revealing symmetry properties related to chain lengths and proving a conjecture, thereby generalizing previous results.

Contribution

It introduces a new adaptation of jeu de taquin for moon polyominoes and proves a conjecture, extending existing combinatorial symmetry results.

Findings

Proved a conjecture of Jakob Jonsson.

Established symmetry properties of fillings.

Connected to Krattenthaler's recent work.

Abstract

We present an adaptation of jeu de taquin and promotion for arbitrary fillings of moon polyominoes. Using this construction we show various symmetry properties of such fillings taking into account the lengths of longest increasing and decreasing chains. In particular, we prove a conjecture of Jakob Jonsson. We also relate our construction to the one recently employed by Christian Krattenthaler, thus generalising his results.