Positive and negative chains in charged moon polyominoes

摘要

We associate with a moon polyomino M a charge function C:M→{±1} such that every cell of the polyomino is either positive or negative. The charge function induces naturally a sign on the northeast and southeast chains of length 2 in 01-fillings of the moon polyomino. We prove that the joint distribution of the numbers of positive chains and negative chains is symmetric and independent of the charge function. Our result reveals a deeper symmetry between the northeast and southeast chains of length 2 in 01-fillings of moon polyominoes, and gives a new family of Mahonian statistics for many combinatorial structures.