Suppose $N$ people are seated in a $K$-sided regular polygon, where each side has $M$ people. What is the number of such arrangement?
The answer given here seems incorrect.
For example, for an equilateral triangle with total $6$ people where each side has $2$ people, the number of arrangement is $\frac{6!}{3}$, as we can rotate three times to get the same combination as initial.
For a square with total $8$ people where each side has $2$ people, the number of arrangement is $\frac{8!}{4}$ due to the same reason above.
Questions:
$(1)$ Do we have a general formula to calculate above arrangemen?
$(2)$ In the two examples given above, it seems that $M$ does not appear in my calculation.
The answer you link to is just counting who is on which side of the polygon, not in what order they sit. In your example, if Alice and Bob sit on the same side, you count it different if Alice is on the left from if Bob is on the left. In the linked problem those are counted the same. You don't need $M$ in your equation, the general form is $\frac {N!}K$ as you surmise. However in the linked answer, the $K!$ in the denominator should just be $K$ as it accounts for rotating the polygon.