Invalid Scenarios of External Cluster Validity Indices: An Analysis Using Bell Polynomial

External cluster validity indices (CVIs) are used to evaluate various clustering algorithms by comparing the obtained clustering result with the gold standard. These indices provide some values for the obtained clustering result, indicating how good or bad the obtained clustering result is compared to the gold standard. For an external CVI, it is desirable to always provide the values within its range; otherwise, the clustering result cannot be correctly interpreted. Thus, in this work, our objective is to theoretically obtain the scenarios where these indices are unable to provide valid values. 26 commonly used indices identified in the existing literature are considered for our analysis purposes. For all these CVIs, we are able to determine the scenarios where these indices provide undefined values. Using the Bell number and Bell polynomial notion, we have attempted to represent the number of such scenarios.