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The inclusion $L((n-1)p)\subseteq L(np)$ allows one additional coefficient: the coefficient of $t^{-n}$ in the local Laurent expansion at $p$, where $t$ is a uniformizer. The resulting linear mapis injective, so the dimension can increase by at most one.

It increases precisely when some rational function has a pole of exact order $n$ at $p$ and no other poles. By the definition of the Weierstrass gap set $G(p)$, this occurs exactly when $n\notin G(p)$. Hence

At every point of a smooth projective curve of genus $g$, exactly $g$ positive integers are Weierstrass gap. Indeed, each quotient $L(np)/L((n-1)p)$ has dimension zero or one, while Riemann--Roch gives $\ell (np)=n+1-g$ for all $n>2g-2$.