Document type
ArticleVersion
Accepted versionPublication date
All rights reserved
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/190456
Alternating Catalan numbers and cover with triple ramification
Journal Title
Director/Tutor
Journal ISSN
Volume Title
Related resource
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 $C_n$.
Subject (English)
Citation
Citation
FARKAS, Gavril, et al. Alternating Catalan numbers and cover with triple ramification. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. 2021. Vol. XXII, num. 2, pags. 665-690. ISSN 0391-173X. [consulted: 16 of August of 2026]. Available at: https://hdl.handle.net/2445/190456