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.
@article{10_37236_3291,
author = {Mitch Phillipson and Catherine H. Yan and Jean Yeh},
title = {Chains of length 2 in fillings of layer polyominoes},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {3},
doi = {10.37236/3291},
zbl = {1295.05035},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3291/}
}
TY - JOUR
AU - Mitch Phillipson
AU - Catherine H. Yan
AU - Jean Yeh
TI - Chains of length 2 in fillings of layer polyominoes
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/3291/
DO - 10.37236/3291
ID - 10_37236_3291
ER -
%0 Journal Article
%A Mitch Phillipson
%A Catherine H. Yan
%A Jean Yeh
%T Chains of length 2 in fillings of layer polyominoes
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/3291/
%R 10.37236/3291
%F 10_37236_3291
Mitch Phillipson; Catherine H. Yan; Jean Yeh. Chains of length 2 in fillings of layer polyominoes. The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/3291
