Pattern Avoidance in Generalized Non-crossing Trees

TL;DR

This paper investigates pattern avoidance in generalized non-crossing trees, deriving generating functions and explicit formulas, and establishing bijections with Schr"{o}der paths, advancing combinatorial understanding of these structures.

Contribution

It introduces new generating functions and explicit formulas for pattern-avoiding generalized non-crossing trees, and establishes a novel bijection with Schr"{o}der paths.

Findings

Generated functions for pattern-avoiding trees

Explicit formulas for special cases

Bijection with Schr"{o}der paths

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 Schr\"{o}der paths.