Részpermutációk

Részpermutációk, often translated as partial permutations, are a concept in combinatorics and abstract algebra. A partial permutation of a set S is a bijective mapping from a subset of S to another subset of S. Unlike a full permutation, which maps the entire set to itself, a partial permutation only operates on a portion of the set.

Formally, a partial permutation $\pi$ on a set $S$ is defined by a pair of subsets $A, B \subseteq S$ and a bijection $\pi: A \rightarrow B$. The domain of the partial permutation is $A$, and its codomain is $B$. The elements of $S \setminus A$ are mapped to themselves, or are considered fixed points, though this definition can vary. Another common definition views a partial permutation as an injective function from a subset of $S$ to $S$. In this case, if the image of the subset is not $S$ itself, it's a proper partial permutation.

Partial permutations are fundamental in various areas. They appear in the study of group theory, particularly in the context of symmetric groups and their subgroups. In computer science, they are relevant in algorithms dealing with rearrangements and data structures. The study of partial permutations also extends to areas like enumeration and the analysis of combinatorial designs. The number of partial permutations of length $k$ from a set of size $n$ is given by $P(n, k) = \frac{n!}{(n-k)!}$, which represents the number of ways to choose and order $k$ elements from a set of $n$.