# Alternating Catalan numbers and curves with triple ramification

## TL;DR

This paper investigates the enumeration of minimal degree covers of general curves with alternating monodromy groups, extending classical results related to symmetric groups and Catalan numbers to the alternating case.

## Contribution

It introduces the problem of counting minimal degree covers with alternating monodromy and provides a solution for the case of genus g curves, expanding the classical Catalan number framework.

## Key findings

- Determined the number of alternating covers of minimal degree 2g+1 for general curves of genus g.
- Extended classical Catalan number results to the case of alternating groups.
- Connected monodromy group classification with enumerative geometry of covers.

## Abstract

It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.