关于不可约映射与切片
On irreducible maps and slices
摘要：
本文考虑d-不可约映射的计数问题，即所有圈的长度至少为d的平面映射，且任何长度为d的圈都是d次面的边界。我们开发了两种方法并行：自然的方法，通过替代这些地图是从一般的地图通过更换所有的d-周期的基本面，和双射的方法，通过切片分解，其中包括在切割的地图沿最短路径。两者都导致显式表达式的生成函数的d-不可约映射与控制面度，总结在一些优雅的“指向公式”。我们提供了一个等价的描述的d-不可约片在所谓的d-导向树。最后，我们表明，不可约映射产生的离散可积方程，其中包括以前遇到的自然嵌入树的上下文中的方程的层次。
英文摘要：
We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the natural approach via substitution where these maps are obtained from general maps by a replacement of all d-cycles by elementary faces, and a bijective approach via slice decomposition which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions of d-irreducible maps with controlled face degrees, summarized in some elegant "pointing formula". We provide an equivalent description of d-irreducible slices in terms of so-called d-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.
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