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 n rows and m 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 S⊆{1,2,...,n} and R(S)
Nearby in the stack
be the union of rows whose indices are in
S
. For any filling
M
, the top-mixed (resp. bottom-mixed) statistic
α(S;M)
(resp.
β(S;M)
) is the sum of the number of northeast chains whose top (resp. bottom) cell is in
R(S)
, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of
R(S)
. Similarly, we define the left-mixed and right-mixed statistics
γ(T;M)
and
δ(T;M)
, where
T
is a subset of the column index set
{1,2,...,m}
. Let
λ(A;M)
be any of these four statistics
α(S;M)
,
β(S;M)
,
γ(T;M)
and
δ(T;M)
, we show that the joint distribution of the pair
(λ(A;M),λ(Aˉ;M))
is symmetric and independent of the subsets
S,T
. In particular, the pair of statistics
(λ(A;M),λ(Aˉ;M))
is equidistributed with
((M),≠(M))
, where
(M)
and
=(M)
are the numbers of southeast chains and northeast chains of