Mixed Statistics on 01-Fillings of Moon Polyominoes

Abstract

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let $\M$ be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of $\M$ in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of $\M$. More precisely, let $S\subseteq \{1,2,...,n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. 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 $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma (T;M)$ and $\delta (T;M)$, where $T$ is a subset of the column index set $\{1,2,...,m\}$. 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 $(\lambda (A;M),\lambda (\bar{A};M))$ is symmetric and independent of the subsets $S,T$. In particular, the pair of statistics $(\lambda (A;M),\lambda (\bar{A};M))$ is equidistributed with $(\se(M),\ne(M))$, where $\se(M)$ and $\ne (M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.

20 pages, 6 figures