2009/08/01 by William Y. C. Chen, Chen, William Y. C., Andrew Y. Z. Wang +5
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0908.0073
20 pages, 6 figures
arxiv created 2009/08/01 · openalex publication_date 2009/08/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ⊆ \1,2,..., n\ and R(S) be the union of rows whose indices are in S. For any filling M, the top-mixed (resp. bottom-mixed) statistic α(S; M) (resp. β(S; M)) is the sum of the number of northeast chains whose top (resp. bottom) cell is in R(S), together with the number of southeast chains whose top (resp. bottom) cell is in the complement of R(S). Similarly, we define the left-mixed and right-mixed statistics γ(T; M) and δ(T; M), where T is a subset of the column index set \1,2,..., m\. Let λ(A; M) be any of these four statistics α(S; M), β(S; M), γ(T; M) and δ(T; M), we show that the joint distribution of the pair (λ(A; M), λ( A; M)) is symmetric and independent of the subsets S, T. In particular, the pair of statistics (λ(A;M), λ( A; M)) is equidistributed with (\se(M),≠(M)), where \se(M) and ≠(M) are the numbers of southeast chains and northeast chains of M, respectively.