<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, we focus on shape partitions. We show that for any fixed <italic>k</italic>, one can symbolically characterize the shape partition on a <italic>k</italic> × <italic>n</italic> rectangular grid by a context-free grammar. We explicitly give this grammar for <italic>k</italic> = 2 and <italic>k</italic> = 3 (for <italic>k</italic> = 1, this corresponds to compositions of integers). From these grammars, we deduce the number of shape partitions for the <italic>k</italic> × <italic>n</italic> rectangular grids for <italic>k</italic> ∈ {1, 2, 3} and every <italic>n</italic>, 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.</p>
</abstract>

In 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.

On the strange kinetic aesthetic of rectangular shape partitions

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On the strange kinetic aesthetic of rectangular shape partitions"}

In 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...

On the strange kinetic aesthetic of rectangular shape partitions

Online veröffentlicht: 18. Juni 2022

Seitenbereich: 37 - 44

Eingereicht: 31. März 2022

Akzeptiert: 15. Mai 2022

DOI: https://doi.org/10.2478/puma-2022-0007

Schlüsselwörter
analytic combinatorics, random sampling

© 2022 Olivier Bodini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the strange kinetic aesthetic of rectangular shape partitions

Online veröffentlicht: 18. Juni 2022

Seitenbereich: 37 - 44

Eingereicht: 31. März 2022

Akzeptiert: 15. Mai 2022

DOI: https://doi.org/10.2478/puma-2022-0007

Schlüsselwörteranalytic combinatorics, random sampling

© 2022 Olivier Bodini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Schlüsselwörter
analytic combinatorics, random sampling