Maximal snake polyominoes are difficult to study numerically in large rectangles, as computing them requires the complete enumeration of all snakes for a specific rectangle size, which corresponds to a brute force algorithm. This hinders the study of maximal snakes in larger rectangles. Moreover, most enumerable snakes lie in small rectangles, obscuring large-scale patterns. In this paper, we investigate the contribution of a deep neural network to the generation of maximal snake polyominoes from a data-driven training, where the maximality and adjacency constraints are not encoded explicitly, but learned. To this extent, we experiment with a denoising diffusion model, which we referred as Structured Pixel Space Diffusion (SPS Diffusion). We find that SPS Diffusion generalizes from small rectangles to larger ones, generating valid snakes up to 28x28 squares and producing maximal snake candidates on squares close to the current computational limit. The model is, however, prone to errors such as branching, cycles, or multiple snake components. Overall, the diffusion model is promising and suggests that complex combinatorial objects can be understood by deep neural networks, which is useful in their investigation.

翻译：最大蛇形多联骨牌在大尺寸矩形中的数值研究十分困难，因为计算该类结构需要对特定尺寸矩形内的所有蛇形结构进行穷举枚举，这本质上属于暴力搜索算法。这种特性阻碍了在更大矩形中研究最大蛇形结构。此外，绝大多数可枚举的蛇形结构均存在于小尺寸矩形中，使得大规模规律难以显现。本文探究了深度神经网络在基于数据驱动的训练过程中生成最大蛇形多联骨牌的贡献——其中最大性约束与邻接约束并非显式编码，而是通过学习获得。为此，我们实验了一种名为结构化像素空间扩散（SPS Diffusion）的去噪扩散模型。研究发现，SPS Diffusion能从较小矩形泛化至较大矩形，生成有效覆盖28×28网格的蛇形结构，并在接近当前计算极限的矩形中产生最大蛇形候选结构。然而该模型仍易出现分支、循环或多蛇形组件等错误。总体而言，该扩散模型展现出良好前景，表明深度神经网络能够理解复杂组合对象，这对相关领域的研究具有重要价值。