Counting fixed points free vector fields on $\mathbb{B}^{2}$

TL;DR

This paper counts the number of stationary point free vector field diagrams on a 2-disk with boundary points, providing a formula involving Catalan numbers and discussing an algorithm for enumeration.

Contribution

It introduces a formula for counting fixed point free vector field diagrams on a disk with boundary points, linking to Catalan numbers and presenting an enumeration algorithm.

Findings

Number of diagrams with 2k boundary points is 3^{k-2}(C_k + 2C_{k-1})

Provides an explicit enumeration formula involving Catalan numbers

Discusses an algorithm for generating all such diagrams

Abstract

The number of diagrams of stationary points free vector fields in the 2-disk $\mathbb{B}^{2}$ is counted in the article. It is shown that the number of such diagrams with $2k$ exceptional points on the boundary $\mathbb{S}^{1}$ equals $3^{k-2}(C_{k}+2C_{k-1})$, where $C_{k}$ is the corresponding Catalan number. An algorithm for finding all such diagrams is discussed.