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 - referred to in other sources as 3-designs. 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.

configuration, configurations of higher orders.

Received: August 4, 2022; Accepted: September 9, 2022; Published: December 5, 2022

How to cite this article: Benjamin Peet, Configurations of higher orders, Advances and Applications in Discrete Mathematics 36 (2023), 11-33. http://dx.doi.org/10.17654/0974165823002

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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