Canalization is a key organizing principle in complex systems, particularly in gene regulatory networks. It describes how certain input variables exert dominant control over a function's output, thereby imposing hierarchical structure and conferring robustness to perturbations. Degeneracy, in contrast, captures redundancy among input variables and reflects the complete dominance of some variables by others. Both properties influence the stability and dynamics of discrete dynamical systems, yet their combinatorial underpinnings remain incompletely understood. Here, we derive recursive formulas for counting Boolean functions with prescribed numbers of essential variables and given canalizing properties. In particular, we determine the number of non-degenerate canalizing Boolean functions -- that is, functions for which all variables are essential and at least one variable is canalizing. Our approach extends earlier enumeration results on canalizing and nested canalizing functions. It provides a rigorous foundation for quantifying how frequently canalization occurs among random Boolean functions and for assessing its pronounced over-representation in biological network models, where it contributes to both robustness and to the emergence of distinct regulatory roles.

翻译：渠道化是复杂系统（尤其是基因调控网络）中的关键组织原则。它描述了某些输入变量如何对函数输出施加主导控制，从而强加层次结构并赋予系统对扰动的鲁棒性。相比之下，退化性捕捉了输入变量之间的冗余性，反映了某些变量被其他变量完全主导的现象。这两种特性都会影响离散动力系统的稳定性和动力学行为，但其组合基础仍未得到充分理解。本文推导了计算具有指定数量本质变量和给定渠道化特性的布尔函数的递归公式。特别地，我们确定了非退化渠道化布尔函数的数量——即所有变量均为本质变量且至少有一个变量具有渠道化特性的函数。我们的方法扩展了先前关于渠道化函数和嵌套渠道化函数的枚举结果。这为量化渠道化在随机布尔函数中出现的频率提供了严格基础，并可用于评估其在生物网络模型中的显著过表达现象——这种过表达既有助于系统鲁棒性，也促进了不同调控角色的形成。