Abstract
Combinatorics The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.
Highlights
A non-crossing tree (NC-tree for short) is a tree drawn on n vertices in {1, 2, · · ·, n} arranged in counterclockwise order along a circle such that the edges lie entirely within the circle and do not cross
Noncrossing trees have been investigated by Chen and Yan [1], Deutsch and Noy [3], Flajolet and Noy [4], Gu, et al [5], Hough [6], Noy [7], Panholzer and Prodinger [8]
Some problems of pattern avoidance in NC-trees have been studied by Sun and Wang [14]
Summary
A non-crossing tree (NC-tree for short) is a tree drawn on n vertices in {1, 2, · · · , n} arranged in counterclockwise order along a circle such that the edges lie entirely within the circle and do not cross. Let GNCn denote the set of rooted GNC-trees of n + 1 vertices. For any σ ∈ Pk, let GNCm n (σ) denote the set of GNC-trees in GNCn which contain the pattern σ exactly m times. For any nonempty subset P ⊂ Pk, GNCn(P ) denotes the set of GNC-trees in GNCn which avoid all the patterns in P. A counterpart for GNC-trees to the analogous question for restricted permutations is the following question: Question 1.2 Determine the cardinalities of GNCn(P ) for P ⊂ Pk and GNCm n (σ) for σ ∈ Pk. In the literature, two kinds of special NC-trees have been considered, namely non-crossing increasing trees and non-crossing alternating trees. A bijection is established between GNC-trees with special pattern avoidance and little Schroder paths
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