zonpathdiagram

A zonpath diagram is a visual representation used in combinatorics and discrete geometry. It depicts the paths on a grid that are confined within a specific region, often a zonotope. A zonotope is a convex polytope that can be expressed as the Minkowski sum of line segments.

The paths within a zonpath diagram are typically restricted to move only in certain directions, such as along the axes of the grid. These paths start at one vertex of the zonotope and end at another, traversing a sequence of edges. The term "zonpath" itself is derived from "zone," referring to the way these paths can be seen as traveling through different "zones" or layers defined by the structure of the zonotope.

Zonpath diagrams are particularly useful for studying combinatorial properties related to hyperplane arrangements and their associated polytopes. They can be employed to count certain types of arrangements or to understand the geometric structure of these objects. The enumeration of zonpaths can often be related to coefficients in polynomial expansions or to other combinatorial quantities. Their study has connections to fields such as algebraic geometry and the theory of Schubert calculus.