Irreducibility of configurations

TL;DR

This paper generalizes the concept of irreducibility in configurations, providing new results on their reducibility and augmentability, which aids in the enumeration and classification of combinatorial configurations.

Contribution

It extends Boben's irreducibility concept to balanced and unbalanced configurations, offering new theoretical insights and criteria for reducibility.

Findings

Generalization of irreducibility for $(v_r,b_k)$-configurations

Identification of conditions for augmentability and reducibility

Results that facilitate counting and classifying configurations

Abstract

In a paper from 1886, Martinetti enumerated small $v_3$-configurations. One of his tools was a construction that permits to produce a $(v+1)_3$-configuration from a $v_3$-configuration. He called configurations that were not constructible in this way irreducible configurations. According to his definition, the irreducible configurations are Pappus' configuration and four infinite families of configurations. In 2005, Boben defined a simpler and more general definition of irreducibility, for which only two $v_3$-configurations, the Fano plane and Pappus' configuration, remained irreducible. The present article gives a generalization of Boben's reduction for both balanced and unbalanced $(v_r,b_k)$-configurations, and proves several general results on augmentability and reducibility. Motivation for this work is found, for example, in the counting and enumeration of configurations.