On the Boolean domain, there is a class of symmetric signatures called ``Fibonacci gates" for which a beautiful P-time combinatorial algorithm has been designed for the corresponding $\operatorname{Holant}^*$ problems. In this work, we give a combinatorial view for $\operatorname{Holant}^*(\mathcal{F})$ problems on a domain of size 3 where $\mathcal{F}$ is a set of arity 3 functions with inputs taking values on the domain of size 3 and the functions share some common properties. The combinatorial view can also be extended to the domain of size 4. Specifically, we extend the definition of "Fibonacci gates" to the domain of size 3 and the domain of size 4. Moreover, we give the corresponding combinatorial algorithms.

翻译：在布尔域上，存在一类称为“Fibonacci门”的对称签名，针对相应的$\operatorname{Holant}^*$问题已设计出优美的多项式时间组合算法。本文给出大小为3的域上$\operatorname{Holant}^*(\mathcal{F})$问题的组合视角，其中$\mathcal{F}$是一组输入取值于大小为3的域、具有公共性质的3元函数集合。该组合视角也可推广至大小为4的域。具体而言，我们将“Fibonacci门”的定义扩展至大小为3的域和大小为4的域，并给出相应的组合算法。