On the symmetry of ascents and descents over 01-fillings of moon polyominoes

TL;DR

This paper explores the symmetry in the distribution of crossings and nestings in certain combinatorial structures, connecting recent results to the enumeration of specific chains in moon polyomino fillings.

Contribution

It extends the understanding of symmetry properties from matchings and partitions to the enumeration of increasing and decreasing chains in moon polyomino fillings.

Findings

Establishes symmetry between crossings and nestings in moon polyomino fillings

Links recent results on set partitions to chain enumeration in polyominoes

Provides a broader combinatorial context for these symmetry phenomena

Abstract

The purpose of this short paper is to put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes.