  The adjusted Rand index (ARI) is commonly used in cluster analysis to measure
the degree of agreement between two data partitions. Since its introduction,
exploring the situations of extreme agreement and disagreement under different
circumstances has been a subject of interest, in order to achieve a better
understanding of this index. Here, an explicit formula for the lowest possible
value of the ARI for two clusterings of given sizes is shown, and moreover a
specific pair of clusterings achieving such a bound is provided.
