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On the strange kinetic aesthetic of rectangular shape partitions

On the strange kinetic aesthetic of rectangular shape partitions AbstractIn this paper, we focus on shape partitions. We show that for any fixed k, one can symbolically characterize the shape partition on a k × n rectangular grid by a context-free grammar. We explicitly give this grammar for k = 2 and k = 3 (for k = 1, this corresponds to compositions of integers). From these grammars, we deduce the number of shape partitions for the k × n rectangular grids for k ∈ {1, 2, 3} and every n, as well as the limiting Gaussian distribution of the number of connected components. This also enables us to randomly and uniformly generate shape partitions of large size. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Pure Mathematics and Applications de Gruyter

On the strange kinetic aesthetic of rectangular shape partitions

Bodini, Olivier
Pure Mathematics and Applications , Volume 30 (1): 8 – Jun 1, 2022
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Publisher
de Gruyter
Copyright
© 2022 Olivier Bodini, published by Sciendo
eISSN
1788-800X
DOI
10.2478/puma-2022-0007
Publisher site
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Abstract

AbstractIn this paper, we focus on shape partitions. We show that for any fixed k, one can symbolically characterize the shape partition on a k × n rectangular grid by a context-free grammar. We explicitly give this grammar for k = 2 and k = 3 (for k = 1, this corresponds to compositions of integers). From these grammars, we deduce the number of shape partitions for the k × n rectangular grids for k ∈ {1, 2, 3} and every n, as well as the limiting Gaussian distribution of the number of connected components. This also enables us to randomly and uniformly generate shape partitions of large size.

Journal

Pure Mathematics and Applications – de Gruyter

Published: Jun 1, 2022

Keywords: analytic combinatorics; random sampling; 05A05; 05A16

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