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In this paper, motivated by the concepts of increasing (alternating) non-crossing trees, the problem of consecutive pattern avoidances in non-crossing trees is proposed. Some given patterns of length two and three are investigated in detail. The Lagrange inversion formula is used to obtain the explicit formulas for these cases. Bijections are established between non-crossing trees avoiding special patterns and Schröder paths.  相似文献
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Yidong Sun 《Discrete Mathematics》2009,309(9):2640-2648
A Motzkin path of length n is a lattice path from (0,0) to (n,0) in the plane integer lattice Z×Z consisting of horizontal-steps (1,0), up-steps (1,1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: “number of u-segments” and “number of h-segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.  相似文献
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