partitities

Partitities refers to a family of quantitative measures associated with partitions of a finite set, used to capture aspects of the partition's granularity and structure. The term is not standardized and different authors define partitities in different ways. Two common definitions are as follows:

- Intra-block partitity P_in(P) = sum_{B in P} (|B| - 1) = |X| - number_of_blocks(P). This measures how many elements remain to be connected within blocks, and decreases as a partition is refined.

- Cross-block partitity P_cross(P) = total number of cross-block pairs = C(|X|,2) - sum_{B in P} C(|B|,2). This reflects how many pairs of elements lie in different blocks, increasing with refinement.

Properties: For a finite set X of size n and a partition P, 0 <= P_in(P) <= n-1, and 0 <= P_cross(P) <= n(n-1)/2. If P is coarser than Q (Q refines P), then P_in(Q) <= P_in(P) and P_cross(Q) >= P_cross(P).

Examples: Let X = {a,b,c,d}. For P = { {a,b}, {c,d} }, P_in(P) = 2 and P_cross(P) = 4. For the coarse partition P' = { {a,b,c,d} }, P_in(P') = 3 and P_cross(P') = 0.

Use and context: Partitities are used as simple, interpretable descriptors of partitions, complementary to more information-theoretic measures like mutual information or the Rand index, and appear in studies of partition lattices and clustering evaluation.

See also: partitions, partition lattice, refinement, equivalence relation.