AI 中文总结
该研究针对具有有界面度的二分图，结合KP层级与Virasoro约束方程构建微分代数系统，得到可计算对应图数量的递推公式，解决了控制二分图面度的计数问题。
AI 中文摘要
近年来，可积层级已被极大地用于组合图的计数，它们导出了关于图的大小和亏格的递推公式，例如三角剖分、二分四角剖分、二分图以及星座图的计数公式。这些公式不仅极为简洁，还是计算这些图数量的最快方法。除了Louf关于星座图的工作外，获得能控制图的面度的递推公式仍是一项挑战。本文展示了如何针对具有有界面度的二分图实现这一点，通过结合KP层级和Virasoro约束的方程，得到了一个微分代数系统，该系统耦合了具有有界根面度的二分图的生成函数，同时控制了边数、黑顶点数、白顶点数以及各度数面的数量（尤其是亏格）。最终，该常微分方程组被证明可给出递推公式，用于计算所有对应的图数量。
英文摘要
In recent years, integrable hierarchies have been used to great advantage for the enumeration of combinatorial maps. They have led to recurrence formulas with respect to the size and genus of the maps, e.g. for triangulations, bipartite quadrangulations and bipartite maps, and for constellations. These formulas are not only remarkably simple but also provide the fastest way of calculating these numbers of maps. With the exception of Louf's work on constellations, it has however remained a challenge to obtain recurrence formulas that control the degrees of the faces of the maps. Here we show how to achieve this for bipartite maps with bounded face degrees. By combining equations from the KP hierarchy and from the Virasoro constraints, a differentially algebraic system is obtained. It couples the generating functions of bipartite maps with bounded root face degrees while controlling the numbers of edges, black vertices, white vertices and number of faces of each degree (and in particular the genus). Finally, this system of ODEs is shown to give recurrence formulas that allows to calculate all the corresponding numbers of maps.
Comments24 pages
