<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

Published Online: Jun 18, 2022

Page range: 37 - 44

Received: Mar 31, 2022

Accepted: May 15, 2022

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

Keywords
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

Published Online: Jun 18, 2022

Page range: 37 - 44

Received: Mar 31, 2022

Accepted: May 15, 2022

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

Keywordsanalytic combinatorics, random sampling

© 2022 Olivier Bodini, published by Sciendo

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

Keywords
analytic combinatorics, random sampling