I am having a plane in N dimension. Th distance between 2 points (a1,a2,...,aN) and (b1,b2,...,bN) is max{|a1-b1|, |a2-b2|, ..., |aN-bN|}.
I need to to know how many K-sets exist(here K-set refers to set of points whose distance between 2 points of set is K).But as there can be infinite number of these K-sets. Thus, we would only like to count the number of classes of K-sets, such that any two K-sets which belong to the same class are equivalent if they follow given conditions.Two K-sets X and Y are considered equivalent (and belong to the same class) if:
They contain the same number of points
There exists N integer numbers (t1, ..., tN) such that by translating each point of X
by the amount ti in dimension i (1≤i≤N) we obtain the set of points Y.
Let's consider N=2, K=4 and the following sets of points X={(1,2), (5,5), (4,3)} and Y={(2,5), (5,6), (6,8)}. Let's consider now the tuple (1,3). By translating each point of X by the amounts specified by this tuple we obtain the set {(2,5), (6,8), (5,6)}, which is exactly the set Y. Thus, the two sets X and Y are equivalent and belong to the same class.
Example let say N=2 and K=1 .
There are 9 classes of K-sets. One K-set from each class is given below:
{(0,0), (0,1)}
{(0,0), (1,0)}
{(0,0), (1,1)}
{(0,1), (1,0)}
{(0,0), (0,1), (1,0)}
{(0,0), (0,1), (1,1)}
{(0,0), (1,0), (1,1)}
{(0,1), (1,0), (1,1)}
{(0,0), (0,1), (1,0), (1,1)}
So answer here will be 9.