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The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm, also called plain changes, is an algorithm named after Hugo Steinhaus, Selmer M. Johnson and Hale F. Trotter that generates all of the permutations of elements. Each two adjacent permutations in the resulting sequence differ by swapping two adjacent permuted elements.
Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way. Explained separately in a more accessible way: Combination. Permutation. For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation)
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 ...
Affine combination. In mathematics, an affine combination of x1, ..., xn is a linear combination. such that. Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K.
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to ...
Combination. In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple ...
Stars and bars (combinatorics) In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", [ 1 ] "balls and bars", [ 2 ] and "dots and dividers" [ 3 ]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems, such as how many ways there are to put n ...
In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of a set of size n is known as the subfactorial of n or the n- th derangement number or n- th de Montmort ...