Mixed Statistics on 01-Fillings of Moon Polyominoes

TL;DR

This paper introduces new symmetric mixed statistics for 01-fillings of moon polyominoes, revealing their joint distributions are symmetric and equidistributed with classical chain counts, thus strengthening the understanding of chain symmetries.

Contribution

The paper establishes a stronger symmetry between northeast and southeast chain counts in 01-fillings of moon polyominoes through novel mixed statistics.

Findings

Joint distribution of mixed statistics is symmetric.

Mixed statistics are independent of subset choices.

Pair of mixed statistics is equidistributed with chain counts.

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.