Mixed Statistics on 01-Fillings of Moon Polyominoes

Paper

Abstract

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let

be a moon polyomino with

rows and

columns. Consider all the 01-fillings of

in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of

. More precisely, let

and

be the union of rows whose indices are in

. For any filling

, the top-mixed (resp. bottom-mixed) statistic

(resp.

) is the sum of the number of northeast chains whose top (resp. bottom) cell is in

, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of

. Similarly, we define the left-mixed and right-mixed statistics

and

, where

is a subset of the column index set

. Let

be any of these four statistics

and

, we show that the joint distribution of the pair

is symmetric and independent of the subsets

. In particular, the pair of statistics

is equidistributed with

, where

and

are the numbers of southeast chains and northeast chains of

, respectively.