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- W1511502611 abstract "The lattice polynomials $L_{i,j}(x)$ are introduced by Hough and Shapiro as a weighted count of certain lattice paths from the origin to the point $(i,j)$. In particular, $L_{2n, n}(x)$ reduces to the generating function of the numbers $T_{n,k}={1over n}{n-1+kchoose n-1}{2n-kchoose n+1}$, which can be viewed as a refinement of the $3$-Catalan numbers $T_n=frac{1}{2n+1}{3nchoose n}$. In this paper, we establish a correspondence between $12312$-avoiding partial matchings and lattice paths, and we show that the weighted count of such partial matchings with respect to the number of crossings in a more general sense coincides with the lattice polynomials $L_{i,j}(x)$. We also introduce a statistic on even trees, called the $r$-index, and show that the number of even trees with $2n$ edges and with $r$-index $k$ equal to $T_{n,k}$." @default.
- W1511502611 created "2016-06-24" @default.
- W1511502611 creator A5039654255 @default.
- W1511502611 creator A5074795333 @default.
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- W1511502611 date "2010-11-16" @default.
- W1511502611 modified "2023-09-27" @default.
- W1511502611 title "Lattice Polynomials, 12312-Avoiding Partial Matchings and Even Trees" @default.
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