Alternating Catalan numbers and cover with triple ramification

Abstract

The Catalan numbers $C_n:=\frac{1}{n+1}\left(\begin{array}{c}2 n \\ n\end{array}\right)$ form one of the most ubiquitous sequence in classical combinatorics. Stanley's book [St] lists 66 different manifestations of these numbers in various counting problems. In the theory of algebraic curves, the Catalan number $C_n$ counts the covers $C \rightarrow \mathbb{P}^1$ of minimal degree $n+1$ from a general curve $C$ of genus $2 n$. Each such cover has simple ramification and its monodromy group equals $S_{n+1}$. By degenerating $C$ to a rational $g$-nodal curve, it was already known to Castelnuovo $[\mathrm{C}]$ that the number of such covers coincides with the degree of the Grassmannian $G(2, n+2)$ in its Plücker embedding, which is well-known to equal $C_n$.