Partition of a Polygon

There is a $n$ -sided polygon $A_1A_2A_3\cdots A_n\,,\,n \geq 4\,,\,$ with the side length as $A_iA_{i+1} = i\,,\, 1 \leq i \leq n \,,\, i \in Z$ and $A_{n+1}$ is taken as $A_1$ . Find all possible partition (through the vertex) of the polygon into two parts such that the difference of the sum of lengths of sides in the two parts is minimum. For clarification of question one partition for $n = 4$ and one for $n = 5$ is shown below. In fact only one partition exists for $n = 4$ and $n = 5$ . Let $s_1\,,\, s_2$ and $s_3$ be the number of possible partitions for $n = 1124\,,\, n = 1755$ and $n = 94$ respectively. Find $s_1 + s_2 + s_3$

BONUS : Solve for possible partitions for any $n$ .