# 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 $\mathcal{M}$ 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.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [14 references]

https://hal.inria.fr/hal-01186273
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Monday, August 24, 2015 - 3:46:46 PM
Last modification on : Friday, April 12, 2019 - 2:26:48 PM
Long-term archiving on: : Wednesday, November 25, 2015 - 5:25:31 PM

### File

dmAN0142.pdf
Publisher files allowed on an open archive

### Citation

William Y. C. Chen, Andrew Y. Z. Wang, Catherine H. Yan, Alina F. Y. Zhao. Mixed Statistics on $01$-Fillings of Moon Polyominoes. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.613-624, ⟨10.46298/dmtcs.2845⟩. ⟨hal-01186273⟩

Record views