Schr\"oder Coloring and Applications

TL;DR

This paper introduces bijections and coloring techniques related to Schr"oder numbers, enabling the enumeration of various combinatorial structures like paths, trees, and maps, along with deriving explicit formulas via Bell polynomial identities.

Contribution

It presents new bijections and coloring methods for Schr"oder objects and derives explicit enumeration formulas using Bell polynomial identities.

Findings

Bijections between Schr"oder objects and combinatorial structures

Enumeration formulas for rational Schr"oder paths, trees, and maps

Explicit identities for Schr"oder numbers using Bell polynomials

Abstract

We present several bijections, in terms of combinatorial objects counted by the Schr\"oder numbers, that are then used (via coloring) for the construction and enumeration of rational Schr\"oder paths with integer slope, ordered rooted trees, and simple rooted outerplanar maps. On the other hand, we derive partial Bell polynomial identities for the little and large Schr\"oder numbers, which allow us to obtain explicit enumeration formulas.