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

The main purpose of this paper is to put recent results of Klazar and Noy [10], Kasraoui and Zeng [9], and Chen, Wu and Yan [2], on the enumeration of 2-crossings and 2-nestings in matchings, set partitions and linked partitions in the larger context of enumeration of increasing and decreasing chains in fillings of arrangements of cells. Our work is motivated by the recent paper of Krattenthaler [11] in which results of Chen et al. [3] on the symmetry of the crossing number and nesting number in matchings and set partitions have been extended in a such context.