Where does arrangement collide?

In combinatorics an arrangement is an ordered selection of some or all of a set of distinct items, so that the same items placed in a different order count as different arrangements. Jacob Bernoulli's Ars conjectandi (1713) set out the systematic theory of permutations and combinations, and the modern combinatorial term arrangement for an ordered selection was standardized in later texts such as Riordan's An introduction to combinatorial analysis (1958). Here the referent is a counted configuration of ordered items, not a way of dividing a plane, a set of economic rules, or a reworked piece of music.[1][2]

Unit

One ordered selection of k items drawn from n distinct items

Recovery means

Not applicable

Measured by

Enumerated by counting formulas, the number of k-item arrangements of n items being n!/(n-k)!

In geometry an arrangement is the subdivision of the plane or of a higher-dimensional space produced by a finite collection of lines, curves, or hyperplanes, described by the vertices, edges, and cells it creates. Jakob Steiner's 1826 paper on how lines partition the plane and space counted these pieces for the first time, and Branko Grünbaum's Arrangements and spreads (1972) gave the modern systematic theory. The referent is a geometric structure carved out of space by intersecting lines, not an ordered list of items, a set of economic rules, or a musical reworking.[3][4]

Unit

One collection of lines or hyperplanes together with the cells they cut space into

Recovery means

Not applicable

Measured by

Characterized by counting its vertices, edges, and cells, with exact counts and bounds derived combinatorially

In institutional economics an arrangement is a specific structure of rules and agreements that governs how economic units cooperate or compete, such as a firm, a contract, or a market convention. Lance Davis and Douglass North's Institutional change and American economic growth (1971) defined the institutional arrangement as the concrete unit of organization operating within a broader institutional environment, and Oliver Williamson's The economic institutions of capitalism (1985) developed how such arrangements are chosen to economize on the costs of transacting. The referent is a governance structure among people and organizations, not a counted list, a geometric figure, or a piece of rescored music.[5][6]

Unit

One governance structure, such as a firm, a contract, or a market convention

Recovery means

Not applicable

Measured by

Studied through the attributes of transactions and the observed choice among market, hybrid, and hierarchy forms