Schr\"oder Coloring and Applications

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.