We find the generating function for the contributions of n-cylinder square-tiled surfaces to the Masur–Veech volume of \(\mathcal{H}(2g-2)\). It is a bivariate generalization of the generating function for the total volumes obtained by Sauvaget via intersection theory. Our approach is, however, purely combinatorial. It relies on the study of counting functions for certain families of metric ribbon graphs. Their top-degree terms are polynomials, whose (normalized) coefficients are cardinalities of certain families of metric plane trees. These polynomials are analogues of Kontsevich polynomials that appear as part of his proof of Witten’s conjecture.

我们找到了n圆柱方瓦表面对 Masur-Veech 体积\(\mathcal{H}(2g-2)\)的贡献的生成函数。它是 Sauvaget 通过交集理论获得的总体积生成函数的二元推广。然而，我们的方法纯粹是组合的。它依赖于对某些度量带状图族的计数函数的研究。它们的顶级项是多项式，其（归一化）系数是某些度量平面树族的基数。这些多项式与 Kontsevich 多项式类似，是证明 Witten 猜想的一部分。