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Configurations of higher orders

23 Mar 2022arXiv:2203.12528links table onlyarchive 2025-07-28

Benjamin Peet

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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.

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