Counting the fission trees and nonabelian Hodge graphs (untwisted case)

Any algebraic connection on a vector bundle on a smooth complex algebraic curve determines an irregular class and in turn a fission tree at each puncture. The fission trees are the discrete data classifying the admissible deformation classes. Here we explain how to count the fission trees with given slope and number of leaves, in the untwisted case. This also leads to a clearer picture of the “periodic table” of the atoms that play the role of building blocks in 2d gauge theory.