正亏格地图双变量生成级数有理性之组合证明

Rationality in map and hypermap enumeration by genus

Generating functions for a fixed genus map and hypermap enumeration become rational after a simple explicit change of variables. Their numerators are polynomials with integer coefficients that obey a differential recursion, and denominators are products of powers of explicit linear functions.

Blossoming bijection for bipartite pointed maps and parametric rationality of general maps of any surface

We construct an explicit bijection between bipartite pointed maps of an arbitrary surface $\mathbb{S}$, and specific unicellular blossoming maps of the same surface. Our bijection gives access to the degrees of all the faces, and distances from the pointed vertex in the initial map. The main construction generalizes recent work of the second author which covered the case of an orientable surface. Our bijection gives rise to a first combinatorial proof of a parametric rationality result concerning the bivariate generating series of maps of a given surface with respect to their numbers of faces …

双变量渐进行为 via 随机游走：应用于大亏格地图

Bivariate asymptotics via random walks: application to large genus maps

研究双变量渐进行为，探讨大亏格地图数量的渐近行为，通过线性递推关系方法，提供一般定理以推导其他线性递推关系的渐近行为。

Asymptotic enumeration of labelled graphs with a given genus

We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled graphs of genus g (1-connected, 2-connected, and 3-connected graphs.)

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalising the Heisenberg group of three by three unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral …

高亏格单部分单调Hurwitz数

High genus one part monotone Hurwitz numbers

本文研究高亏格下单部分单调Hurwitz数的双变量渐进行为，通过线性递推关系和Elvey-Price等人提出的方法提取渐近结果。