Predators and altruists arriving on jammed Riviera

TL;DR

The paper extends the Riviera jammed-state combinatorics by introducing predators and altruists as post-jamming builders and develops exact counting tools for jammed configurations resistant to these actors. It defines $J_{k,n}^{P}$ as the number of jammed configurations of length $n$ with $k$ occupied lots that remain resistant to predators, and constructs a transfer-matrix framework with forbidden patterns $111$, $000$, $0100$, $0010$ plus the constraint of two consecutive zeros and boundary occupancy to obtain a $4$-state directed graph; the four vertices correspond to substrings $011$, $110$, $101$, and $010$, and a matrix function yields the associated $bivariate$ generating function. The study also yields analogous counts for maximal configurations resistant to altruists and to both actor types, accompanied by configurational-entropy analyses and connections to the two-dimensional settlement framework. The results provide exact combinatorial tools and entropy measures for assessing resistance properties in jammed settlements, with implications for evolutionary dynamics and governance under adversarial and cooperative pressures.

Abstract

The Riviera model is a combinatorial model for a settlement along a coastline, introduced recently by the authors. Of most interest are the so-called jammed states, where no more houses can be built without violating the condition that every house needs to have free space to at least one of its sides. In this paper, we introduce new agents (predators and altruists) that want to build houses once the settlement is already in the jammed state. Their behavior is governed by a different set of rules, and this allows them to build new houses even though the settlement is jammed. Our main focus is to detect jammed configurations that are resistant to predators, to altruists, and to both predators and altruists. We provide bivariate generating functions, and complexity functions (configurational entropies) for such jammed configurations. We also discuss this problem in the two-dimensional setting of a combinatorial settlement planning model that was also recently introduced by the authors, and of which the Riviera model is just a special case.

Figures (6)

• Figure 1: An example of a tract of land ($m = 5$, $n = 7$).
• Figure 2: Examples of impermissible, permissible and maximal configuration on a $5 \times 4$ tract of land. The houses that are blocked from the sunlight are marked with 'x'.
• Figure 3: An example of a jammed configuration on which a predator could build a house. If a house is built on the lot marked with 'x', then this house will receive sunlight from the east. It will also block the sunlight to the house to its west, but that does not concern the predator. Notice that the predator could, equally, build a house on the lot to the east of the one marked with 'x' (but not on both).
• Figure 4: An example of a jammed configuration on which an altruist could build a house. If a house is built on the lot marked with 'x', then this house will not block sunlight to its east nor its west neighbor, since the one to the west is exposed to sunlight from the west, and the one to the east is exposed to sunlight from the east. The house built on the lot marked with 'x', however, will not receive any sunlight, but that does not concern the altruist.
• Figure 5: Examples of maximal configurations resistant to predators, to altruists, and to both predators and altruists.