Consecutive Pattern Avoidances in Non-crossing Trees

Abstract

In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is used to obtain the explicit formulas for some special cases. Bijection is also established between generalized non-crossing trees with special pattern avoidance and the little Schröder paths.

A Decomposition Algorithm for Noncrossing Trees

“…It is well known that the number of NC-trees of size n equals 1 2n+1 3n n , the generalized Catalan number. Noncrossing configurations have been extensively studied, see, for example, [1,3,[5][6][7][8][9][10][11]13]. In a NC-tree T , a vertex v is planted at u if (u, v) is an edge and u lies on the unique path from the root to v. Moreover, (u, v) is a descent if u > v. Denote N n,k the number of NC-trees of size n with k descents.…”

“…13, 21,12,7,5, 14, 18, 17) belongs to A 11,5 . By definition, we have |A 2k+1,k | = (k + 1)k|P 2k+1,k |.…”