Hello,
We were very happy to see your preprint "Enumerating Seating Arrangements that Obey Social Distancing" on arxiv in which you cited our article "Packing density of combinatorial settlement planning models" from the American Mathematical Monthly. It is always a pleasure to read something new and interesting related to this nice topic.
As we mentioned in that paper, we were introduced to this problem by our friend Juraj Bozic who came up with it during his studies at Faculty of Architecture, University of Zagreb. We considered this model already in the paper Combinatorial settlement planning that appeared recently in Contributions to Discrete Mathematics.
We realized that in your recent preprint you mention a few other problems that we were also working on, so we decided to write a short email about that, in case you find it interesting. We considered one-dimensional version of our problem (without the sun coming from the south) in the preprint On a variant of Flory model . You can see that the function f3(z, x) on page 6 of your preprint is the same function that we have on the top of page 9 in our preprint. Additionally, the corresponding limiting average density of 0.5772029462 is also mentioned in our preprint on the top of page 10.
It turns out that physicists are also very interested in these kinds of models. In particular, the setting in which each person must have at least b empty seats on either side translates to the setting of the so-called Rydberg atoms with blockade range b. We tackled the model of Rydberg atoms on a one-dimensional lattice in the preprint Complexity Function of Jammed Configurations of Rydberg Atoms in which we obtained your conjectured function gb(z, x) (that appears on page 7 of your preprint). In our preprint, the function is called Fb and it appears on the top of page 6. There are a few other preprints, but not so closely related to things that you mention. For example, we recently considered the problem of Rydberg atoms on a ladder (in RSA setting, and in equilibrium setting - see here.
If you will have some additional comments, questions or remarks, feel free to contact us. We are more than happy to discuss these things.