Analytic combinatorics in several variables is a branch of mathematics that deals with deriving the asymptotic behavior of combinatorial quantities by analyzing multivariate generating functions. We study information-theoretic questions about sequences in a discrete noiseless channel under cost constraints. Our main contributions involve the relationship between the graph structure of the channel and the singularities of the bivariate generating function whose coefficients are the number of sequences satisfying the constraints. We use these new results to invoke theorems from multivariate analytic combinatorics to obtain the asymptotic behavior of the number of cost-limited strings that are admissible by the channel. This builds a new bridge between analytic combinatorics in several variables and labeled weighted graphs, bringing a new perspective and a set of powerful results to the literature of cost-constrained channels. Along the way, we show that the cost-constrained channel capacity is determined by a cost-dependent singularity of the bivariate generating function, generalizing Shannon's classical result for unconstrained capacity, and provide a new proof of the equivalence of the combinatorial and probabilistic definitions of the cost-constrained capacity.

翻译：多元解析组合学是数学的一个分支，它通过分析多元生成函数来推导组合量的渐近行为。我们研究了在成本约束下离散无噪声信道中序列的信息论问题。我们的主要贡献涉及信道图结构与二元生成函数奇点之间的关系，该生成函数的系数是满足约束条件的序列数量。利用这些新结果，我们援引多元解析组合学中的定理，得到了信道可容许的成本受限字符串数量的渐近行为。这在多元解析组合学与带标签加权图之间建立了新的桥梁，为成本受限信道的研究文献带来了新的视角和一系列强有力的结果。在此过程中，我们证明了成本受限信道容量由二元生成函数的一个成本相关奇点决定，从而推广了香农关于无约束容量的经典结果，并为成本受限容量的组合定义与概率定义之间的等价性提供了新的证明。