Partial permutation

A partial permutation of a set is an injective function defined on a subset of that set. Equivalently, it is an ordered selection of distinct elements.

Connect

Function: A function assigns each element of a domain exactly one element of a codomain. A partial permutation is a function whose domain may be smaller than the whole set.
Falling factorial: The falling factorial counts ordered selections of a fixed number of distinct objects. For a domain of size k in an n-element set, there are n falling-factorial k choices.
Partial matching: A matching is a set of edges with no shared endpoints; it may leave vertices unmatched. An injective assignment between two sets encodes a matching that need not cover either side.
Combination: A combination is a selection of elements in which order does not matter. A partial permutation distinguishes selections that contain the same elements in different orders.
Injective function: A function is injective when distinct inputs always have distinct outputs. Injectivity is the defining condition that prevents repeated selected elements.
Binomial coefficient: A binomial coefficient counts the ways to choose a fixed-size subset from a finite set. Choosing the k-element domain contributes the factor that combines with the falling factorial.
Rook polynomial: A polynomial whose coefficients count nonattacking rook placements on a board. Each placement records a partial one-to-one assignment between rows and columns.
Variation with repetition: An ordered selection in which the same element may be chosen more than once. Partial permutations forbid the repeated entries allowed in variations with repetition.
Subset: A subset is a set whose elements all belong to another set. The domain of a partial permutation is a subset of the underlying set.
Partial transformation semigroup: A semigroup of all partial functions from a set to itself, composed where defined. Partial permutations form a subsemigroup when partial functions are composed.