# Hyperplanes in Configurations, decompositions, and Pascal Triangle of Configurations

## TL;DR

This paper explores hyperplanes in specific configuration classes, revealing a Pascal-like decomposition process, and introduces new classes of configurations with potential for further research.

## Contribution

It demonstrates that certain configuration decompositions are due to fixed hyperplanes within classes, extending previous work with new examples and open questions.

## Key findings

- Identifies hyperplanes that induce configuration decompositions
- Introduces two new natural classes of configurations
- Proposes open questions for future research

## Abstract

An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented in \cite{perspect}. We show that such a procedure is a result of fixing in configurations in some class $\mathcal K$ suitable hyperplanes which both: are in this class, and deleting such a hyperplane results in a configuration in this class. By a way of example we show two more (added to that of \cite{gevay}) natural classes of such configurations, discuss some other, and propose some open questions that seem also natural in this context.