A note on meromorphic functions on a compact Riemann surface having poles at a single point

On a compact Riemann surface X of genus g, one of the questions is the existence of meromorphic functions having poles at a point P on X. One of the theorems is the Weierstrass gap theorem that determines a sequence of g numbers 1 < nₖ < 2g, 1 ≤ k ≤ g for which a meromorphic function with the order with nₖ fails to exist at P. In this note, we give proof of the Weierstrass gap theorem in cohomology terminology. We see that an interesting combinatorial problem may be formed as a byproduct from the statement of the Weierstrass gap theorem.