Increasing and Decreasing Sequences of Length Two in 01-Fillings of Moon Polyominoes

TL;DR

This paper explores the enumeration of increasing and decreasing chains of length two in moon polyominoes, connecting recent symmetry results in matchings and set partitions to broader combinatorial structures.

Contribution

It extends the understanding of chain enumeration in moon polyominoes by relating it to the symmetry of crossings and nestings in various combinatorial objects.

Findings

Established a connection between chain enumeration and symmetry of crossings and nestings.

Generalized recent symmetry results to moon polyominoes.

Provided enumeration formulas for increasing and decreasing chains of length two.

Abstract

We 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.