Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer–Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to Bousquet-Mélou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

Harer和Zagier证明了递归以枚举2 d- gon的粘合，从而产生了可定向的属g表面，在他们对曲线的模空间的欧拉特性的研究中。对于其他枚举问题，已经发现了类似的结果，因此很自然地提出以下问题：存在这些所谓的1点递归的问题家族有多大？在本文中，我们证明了一类具有Schur函数扩展的枚举问题的1点递归。特别是，我们恢复了Harer-Zagier递归，但是我们的方法也适用于dessins d'enfant枚举，Bousquet-Mélou-Schaeffer数，单调Hurwitz数等等。另一方面，我们证明不存在控制单个Hurwitz数的1点递归。我们的结果在一定程度上可以有效地计算出1点递归的特定实例时是有效的，我们提供了几个示例。我们以简短的讨论和一个将一点递归与拓扑递归理论相关的猜想作为本文的结尾。