On the greatest distance between two partitions of the finite set

Abstract

Consider the family Ξ of all possible partitions of a given finite set into disjoint parts. Suppose we have A ∈ Ξ, and there is reason to consider this partition basic from a certain point of view. The greatest value d *(A) of the special cluster metric d(A, B) is found, which is reached when its second argument runs through all B ∈ Ξ. The value of d *(A) turns out to depend on the structure of the basic partition A. Using the found value of d *(A), we propose a numerical coefficient whose value allows us to estimate the degree of difference between the basic partition and the newly built one. Such an assessment can help us to make a decision about the possibility of using the new partition instead of the basic one.