AI 中文总结
本文针对k=1、k=2情形，利用切片分解与可达性概念，以纯组合方式研究带混合边界的平面超图枚举问题，复现了此前代数方法得到的相关公式。
AI 中文摘要
平面超图是一类赋予内部面黑白二色恰当着色的平面映射。本文研究具有指定面度与k-交替边界条件的平面超图枚举问题：该边界条件指绕超图一周时，与外表面关联的内部面颜色交替次数至多为2k次。本文处理k=1、k=2的情形，一般情形留待后续第II部分讨论。本文方法依赖于所谓的切片分解，并关键利用可达性概念——该概念通过超图的典范定向及与顶点关联的标记变量t，依据所有可到达标记顶点的顶点集合分解带标记超图，从而实现带标记超图的枚举。该过程使我们能将带混合边界的超图生成函数表示为超图切片生成函数的形式，并以纯组合方式复现此前通过代数方法得到的部分公式。
英文摘要
Plane hypermaps are plane maps endowed with a proper coloration of their inner faces in black or white. We consider the problem of enumerating plane hypermaps with prescribed face degrees and a $k$-alternating boundary condition: by this we mean the colors of inner faces incident to the outer face alternates at most $2k$ times when turning around the hypermap. The present paper deals with the cases $k=1,2$, the general case being left to the forthcoming part II. Our approach relies on the so-called slice decomposition and uses crucially the notion of accessibility, which exploits the canonical orientation of hypermaps and the marking variable $t$ associated with vertices, to enumerate pointed hypermaps by decomposing them according to the set of all vertices that can access to the marked vertex. This process enables us to express the generating functions of hypermaps with mixed boundaries in terms of the generating functions of hypermap slices and to recover, in a purely combinatorial way, some formulas previously obtained through algebraic methods.
Comments28 pages, 11 figures
