$d$-集映射的解析组合学及其应用
Analytic Combinatorics of $d$-Set Mappings and Their Applications
- School of Mathematics and Statistics, Carleton University（卡尔顿大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出解析组合框架，利用生成函数和奇点分析量化$d$-集映射函数图的宏观性质，推导渐近展开式，推广经典二分映射结果，为概率极限定律奠定基础。
AI中文摘要：
$d$-集映射是作用于定义域$X$上的函数，该定义域被划分为$d$个不相交子集。标准函数代表$1$-集映射，而对任意$d$-部结构的推广在离散数学中自然出现。本文发展了一个解析组合框架来量化这些映射的函数图。通过利用生成函数和奇点分析，我们推导出当$X$的基数趋于无穷时宏观图性质的精确渐近展开式，包括连通分量、循环节点和尾长的期望数量。我们通过恢复Hansen和Jaworski的经典二分映射结果来展示该框架的有效性，并成功将这些机制推广到任意$d$-集映射，为建立其概率极限定律提供了基础架构。
英文摘要：
A $d$-set mapping is a function acting on a domain $X$ equipped with a partition into $d$ disjoint subsets. While standard functions represent $1$-set mappings, generalizations to arbitrary $d$-partite structures appear naturally across discrete mathematics. In this paper, we develop an analytic combinatorial framework to quantify the functional graphs of these mappings. By leveraging generating functions and singularity analysis, we derive exact asymptotic expansions for macroscopic graph properties as the cardinality of $X$ tends to infinity, including the expected number of connected components, cyclic nodes, and tail lengths. We demonstrate the efficacy of this framework by recovering the classical bipartite mapping results of Hansen and Jaworski, and successfully generalize these mechanisms to arbitrary $d$-set mappings, providing the foundational architecture to establish their probabilistic limit laws.
