论文标题
在平行扩散中计数路径配置
Counting Path Configurations in Parallel Diffusion
论文作者
论文摘要
平行扩散是Duffy等人在2018年引入的芯片发射变体。在平行的扩散中，芯片从高浓度的地方通过离散时间的过程移动到低浓度的地方。在每个时间步骤中，每个顶点都会向每个贫穷的邻居发送一个筹码，从而使某些顶点可能会陷入债务（由负堆栈尺寸表示）。在最近的论文中，朗和纳拉亚南证明了Duffy等人的原始论文的猜想。在经过一定的之前，每个平行扩散过程最终都会表现出周期性的行为。通过此结果，我们现在能够计算出存在于同构定义的这些时期的数量。我们确定用于计算任何长度路径的数字的复发关系。如果$ t_n $是具有周期长度2的配置数，可以在$ p_n $上存在，而同构为同构，而$ n $的整数大于4，我们得出结论，$ t_n = 3t_ = 3t_ {n-1} + 2t_ {n-2 {n-2} + t_ {n-2} + t_ {n-3} {n-3} - t_ {n-3} - t_ {n-4} $ {n-4} $。
Parallel Diffusion is a variant of Chip-Firing introduced in 2018 by Duffy et al. In Parallel Diffusion, chips move from places of high concentration to places of low concentration through a discrete-time process. At each time step, every vertex sends a chip to each of its poorer neighbours, allowing for some vertices to perhaps fall into debt (represented by negative stack sizes). In their recent paper, Long and Narayanan proved a conjecture from the original paper by Duffy et al. that every Parallel Diffusion process eventually, after some pre-period, exhibits periodic behaviour. With this result, we are now able to count the number of these periods that exist up to a definition of isomorphism. We determine a recurrence relation for calculating this number for a path of any length. If $T_n$ is the number of configurations with period length 2 that can exist on $P_n$ up to isomorphism and $n$ is an integer greater than 4, we conclude that $T_n = 3T_{n-1} + 2T_{n-2} + T_{n-3} - T_{n-4}$.
