On the number of clusterings in a hierarchical classication model with overlapping clusters

This paper shows a new combinatorial problem which emerged from studies on an artificial intelligence classification model of a hierarchical classifier. We introduce the notion of proper clustering and show how to count their number in a special case when 3 clusters are allowed. An algorithm that generates all clusterings is given. We also show that the proposed approach can be generalized to any number of clusters, and can be automatized. Finally, we show the relationship between the problem of counting clusterings and the Dedekind problem.