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

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.