For a closed plane curve $C$, define its sequence of winding numbers to be the sorted list of the winding numbers of each of the distinct regions of the plane demarcated by $C$. For example, this curve (if I've calculated correctly) has sequence $001111223 = 0^2 1^4 2^2 3$.
A winding number sequence must include $0$ for the unbounded region of the plane. I am wondering if there are any other restrictions:
Q. Can any winding sequence of consecutive integers that includes $0$ be realized by some curve $C$?