geometry – Combination in symmetry


Let’s assume we have the six identical vertices connected with two different lines.

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I want to count how many unique ways exist to make partition of the system.

For example, if I want to make 1×5 partitions,

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the partition has rotational symmetry with a 60-degree angle (any 1×5 will have one green and one black connection), so I only have one unique partition. However, if I use a combination, then I will have 6C1=6 cases. How do I take into account the symmetry?

For 3×3 partitions, it became a little more complicated, because of 6C3=20 cases. However, when we take account of the symmetry, we only have 4 unique cases. By getting rid of the double counting 6C3/2=10, and we have (1(3green and 3 black)+3(Green+Black)+3(3black and one green)+3(3 green and one black).

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Is there any way that I can find the generalized combination solution with the symmetry of the system?