Authors: William Y. C. Chen ¹; Andrew Y. Z. Wang ¹; Catherine H. Yan ²; Alina F. Y. Zhao ¹

• 1 Center for Combinatorics [Nankai]
• 2 Department of Mathematics [Texas]

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of $01$-fillings of moon polyominoes. Let $\mathcal{M}$ be a moon polyomino. Consider all the $01$-fillings of $\mathcal{M}$ in which every row has at most one $1$. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\mathcal{M}$. More precisely, let $S$ be a subset of rows of $\mathcal{M}$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha (S; M)$ (resp. $\beta (S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $S$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $S$. Similarly, we define the left-mixed and right-mixed statistics $\gamma (T; M)$ and $\delta (T; M)$, where $T$ is a subset of the columns. Let $\lambda (A; M)$ be any of these four statistics $\alpha (S; M)$, $\beta (S; M)$, $\gamma (T; M)$ and $\delta (T; M)$. We show that the joint distribution of the pair $(\lambda (A; M), \lambda (M/A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda (A;M), \lambda (M/A; M))$ is equidistributed with $(\mathrm{se}(M), \mathrm{ne}(M))$, where $\mathrm{se}(M)$ and $\mathrm{ne}(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.