Where does partition collide?

A division of a set into a family of non-empty, non-overlapping pieces whose union is the whole set, so that every element belongs to exactly one piece: such a division corresponds precisely to an equivalence relation, in which the pieces are the classes of elements treated as the same. Rota's 1964 The Number of Partitions of a Set studied these divisions and counted them, giving the Bell numbers that record how many ways a set can be split. The referent is an abstract grouping of a set's members, fixed only by which elements share a block, and it has nothing to do with slicing a storage device, summing over physical states, or dividing a country.[1]

Unit

One partition of a set: a collection of disjoint non-empty blocks that cover every element once

Recovery means

Not applicable

Measured by

Specified by listing its blocks, and counted across all partitions of an n-element set by the Bell numbers

In statistical mechanics, the quantity formed by adding up every accessible state of a system, each weighted by its Boltzmann factor, and usually written Z: once this total is known, the thermodynamic properties of the system, such as its energy, entropy, and pressure, can all be derived from it. Gibbs's 1902 Elementary Principles in Statistical Mechanics set out the ensemble framework in which this sum over states plays its central role. The referent is a single number computed from a system's energy levels and temperature, from which physical quantities follow, and it bears no relation to grouping a set, slicing a drive, or splitting a country.[2]

Unit

A single number Z for a system at a given temperature, the weighted total over all its states

Recovery means

Not applicable

Measured by

Computed by adding the Boltzmann factor of every accessible state, then used to derive energy, entropy, and other properties

In political geography and history, the division of a territory into two or more separately governed units, typically creating a new political border across land that was previously one jurisdiction: the word names both the act of drawing such a border and its outcome. O'Leary's 2007 Analysing partition: Definition, classification and explanation offered a careful definition of the concept and a scheme for classifying and comparing cases. The referent is a redrawing of political boundaries over inhabited land, studied through its causes and consequences, and it shares nothing with a mathematical grouping, a region of a disk, or a sum over physical states.[3]

Unit

One partition: a new political border dividing a territory into separately governed units

Recovery means

Not applicable

Measured by

Identified by the drawing of a new international or internal border, and studied through its causes and effects