  To devise efficient solutions for approximating a mean partition in consensus
clustering, Dimitriadou et al. [3] presented a necessary condition of
optimality for a consensus function based on least square distances. We show
that their result is pivotal for deriving interesting properties of consensus
clustering beyond optimization. For this, we present the necessary condition of
optimality in a slightly stronger form in terms of the Mean Partition Theorem
and extend it to the Expected Partition Theorem. To underpin its versatility,
we show three examples that apply the Mean Partition Theorem: (i) equivalence
of the mean partition and optimal multiple alignment, (ii) construction of
profiles and motifs, and (iii) relationship between consensus clustering and
cluster stability.
