NDLI: Mixed Statistics on 01-Fillings of Moon Polyominoes

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wang, Andrew Y. Z. Yan, Catherine H. Zhao, Alina F. Y. Chen, William Y. C.
Copyright Year	2010
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 with n rows and m columns. 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 $S\subseteq\{1,2,\dots,n\}$ and let $\mathcal{R}(S)$ 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 $\{1,2,\dots,m\}$. 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 $(\mathrm{se}(M),\mathrm{ne}(M))$, where $\mathrm{se}(M)$ and $\mathrm{ne}(M)$ are the numbers of southeast chains and northeast chains of M, respectively.
Starting Page	1272
Ending Page	1290
Page Count	19
File Format	PDF
ISSN	08954801
DOI	10.1137/090769065
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	24
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-10-05
Access Restriction	Subscribed
Subject Keyword	mixed statistic Exact enumeration problems, generating functions symmetric distribution polyomino Partitions of sets Permutations, words, matrices
Content Type	Text
Resource Type	Article
Subject	Mathematics

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