Arrangementsfrom

Arrangementsfrom is a term used in combinatorics and computer science to describe the construction or enumeration of all ordered selections of elements drawn from a given source. It is often encountered as a descriptive name for a function or procedure that generates k-element arrangements from a set or multiset, typically without replacement.

Definition and scope

For a finite source S with size n and an integer k with 0 ≤ k ≤ n, arrangementsfrom(S, k) denotes the collection of all sequences (s1, s2, ..., sk) where each si is an element of S and no element is repeated within a sequence. The number of such arrangements, when S contains n distinct elements, is the k-permutation count P(n, k) = n!/(n−k)!.

Extensions and variations

When the source is a multiset with element multiplicities, the concept extends by accounting for how many times each value can appear and whether identical copies are treated as distinct. In many programming contexts, arrangementsfrom can be implemented in two common modes: without replacement (respecting multiplicities) and with replacement (allowing repetition). With replacement, the number of arrangements is n^k, since each position can be filled by any of the n elements independently.

Algorithms and implementation

Common algorithms for generating arrangementsfrom include backtracking, iterative lexicographic generation, and index-based mappings that convert a rank in 0..P(n,k)−1 into a corresponding k-tuple. When duplicates are present, implementations may either produce distinct sequences considering identical elements as indistinguishable or treat each copy as unique, depending on the chosen convention.

Applications

Arrangementsfrom concepts are used in combinatorial enumeration, testing and sampling strategies, puzzle design, and algorithm benchmarking. In software libraries, arrangementsfrom is often realized as a function that yields or collects all k-permutations from a given collection.

See also

k-permutations, permutations, combinations, multiset permutations, Cartesian product.