An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length n has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient C(n,k) where n is the length and k is the weight of the binary string.

中文翻译：

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一维非均匀数守恒初等元胞自动机规则的构造

从元胞自动机的指数阶规则空间研究一维非均匀基本数守恒元胞自动机 (NCCA) 规则是一项出色的计算任务。为了有效地执行这项任务，在长度为二进制字符串的守恒函数数量下的数学遗产n与它们的数量守恒元胞自动机规则（统一或非统一）一起突出显示。研究了在九个统一基本 CA 规则的帮助下构建任何有限配置的一些可行的非统一 NCCA 规则的基本方法。从我们的构建过程中，递推方程被制定为适当地求解以确定 NCCA 规则的实际范围。分析了NCCA规则的状态转移图（STD）。在通过 STD 对二进制字符串进行分类时，我们发现了一个令人着迷的光学见解，即基数与二项式系数相同的类的权重相等C(n,ķ)在哪里n是长度和ķ是二进制字符串的权重。