Configurations of higher orders

TL;DR

This paper extends the concept of combinatorial configurations to higher orders, exploring configurations of points and k-planes, and analyzing symmetric configurations of order 2 with specific properties.

Contribution

Introduces the notion of configurations of order k, generalizing classical configurations, and investigates their properties, examples, and enumeration for symmetric cases.

Findings

Defined configurations of order 2 and k

Constructed examples like stacked and product configurations

Enumerated symmetric configurations with specific parameters

Abstract

This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further extension to the notion of points and $k$-planes ($k$-dimensional hyperplanes) which we refer to as configurations of order $k$. We present a number of general examples such as stacked configurations of order $k$ - intuitively layering lower order configurations - and product configurations of order $k$. We discuss many analogues of standard configurations such as dual configurations, isomorphisms, graphical representations, and when a configuration is geometric. We focus mostly on configurations of order $2$ and specifically compute the number of possible symmetric configurations of order $2$ when each plane contains $3$ points for small values on $n$ - the total number of points in the configuration.