Chains of Length 2 in Fillings of Layer Polyominoes

Abstract

The symmetry of the joint distribution of the numbers of crossings and nestings of length $2$ has been observed in many  combinatorial structures, including permutations, matchings, set partitions, linked partitions, and certain families of graphs.  These results have been unified in the larger context of enumeration of northeast and southeast chains of length $2$ in $01$-fillings of moon polyominoes. In this paper  we extend this symmetry to fillings of a more general family—layer polyominoes, which are intersection-free and row-convex, but not necessarily column-convex.  Our main result is that the joint distribution of the numbers of northeast  and southeast chains of length $2$ over $01$-fillings is symmetric and invariant under an arbitrary permutation of rows.