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Permutations without repetition on the left, with repetition to their right. If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which each element appears a number of times equal exactly to its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset ...
Some authors restrict partial permutations so that either the domain [4] or the range [3] of the bijection is forced to consist of the first k items in the set of n items being permuted, for some k. In the former case, a partial permutation of length k from an n-set is just a sequence of k terms from the n-set without repetition.
In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n+1 objects.
A requirement that be injective means that no label can be used a second time; the result is a sequence of labels without repetition. In the absence of such a requirement, the terminology "sequences with repetition" is used, meaning that labels may be used more than once (although sequences that happen to be without repetition are also allowed).
The minimum distance of a permutation code is defined to be the minimum positive integer such that there exist ,, distinct, such that (,) =. One of the reasons why permutation codes are suitable for certain channels is that the alphabet symbols only appear once in each codeword, which for example makes the errors occurring in the context of ...
In combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums. [1] Separable permutations may be characterized by the forbidden permutation patterns 2413 and 3142; [2] they are also the permutations whose permutation graphs are cographs and the permutations that realize the series-parallel partial orders.
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation.Any permutation may be written in one-line notation as a sequence of entries representing the result of applying the permutation to the sequence 123...; for instance the sequence 213 represents the permutation on three elements that swaps elements 1 and 2.